archive.ymsc.tsinghua.edu.cnarchive.ymsc.tsinghua.edu.cn/pacm_download/21/1000-JDG...j.differential...

j. differential geometry

95 (2013) 419-481

THIN INSTANTONS IN G2-MANIFOLDSAND SEIBERG-WITTEN INVARIANTS

Naichung Conan Leung, Xiaowei Wang & Ke Zhu

Abstract

For two nearby disjoint coassociative submanifolds C and C′ ina G2-manifold, we construct thin instantons with boundaries lyingon C and C′ from regular J-holomorphic curves in C. We explaintheir relationship with the Seiberg-Witten invariants for C.

1. Introduction

Intersection theory of Lagrangian submanifolds is an essential partof symplectic geometry. By counting the number of holomorphic disksbounding intersecting Lagrangian submanifolds, Floer and others de-fined the celebrated Floer homology theory. It plays an important role inmirror symmetry for Calabi-Yau manifolds and string theory in physics.In M-theory, Calabi-Yau threefolds are replaced by seven dimensionalG2-manifolds M (i.e. oriented Octonion manifolds [26]). The analogsof holomorphic disks (resp. Lagrangian submanifolds) are instantons orassociative submanifolds (resp. coassociative submanifolds or branes) inM [25]. In [17], the Fredholm theory for instantons with coassociativeboundary conditions has been set up. However, existence of instantonsis still a difficult problem. As a first step, we want to give a constructionmodeled on the work of Fukaya and Oh [10] in symplectic geometry. Asit was shown in [10], if we choose two nearby Lagrangian submanifoldsin such a way that one is the graph of a closed one form on the other,then the holomorphic disks bounding two are closely related to gradientflow lines of the one form. Searching for the analog in G2-geometry leadsus to study the following problem.

Problem: Given two nearby coassociative submanifolds C and C ′

in a (almost) G2-manifold M , relate the number of instantons in Mbounding C ∪ C ′ to the Seiberg-Witten invariants of C.

The basic idea is as follows: When the coassociative submanifold C ′ issufficiently close to C, then it is the graph of a self-dual two form on C.This two form is essentially a symplectic form on C away from the inter-section C ∩C ′. Instantons bounding C ∪C ′ would become holomorphic

Received 3/29/2012.

419

420 N.C. LEUNG, X. WANG & K. ZHU

curves on C when C ′ collapses onto C modulo the possible bubblings. Bythe seminal work of Taubes [36] on the equivalence of Gromov-Wittenand Seiberg-Witten invariants, we expect that the number counted withalgebraic weights of such instantons is given by the Seiberg-Witten in-variants of C.

Settling the above problem completely is very difficult at the currentstage. We treat the special case when C and C ′ are disjoint, i.e. C isa symplectic four manifold in this paper. The basic tool is the gluingtechnique. But even in this simpler case, the set up is quite differentfrom the Lagrangian Floer theory (cf. Fukaya and Oh [10]). Our do-mains are three dimensional instead of two dimensional, and we have todeform the submanifolds rather than deforming the maps as was donein Floer theory. This is because we do not have the luxury of applyingthe conformal geometry in dimension two to transform the problem offinding holomorphic curves into the one of finding holomorphic maps.Furthermore, the linear theory is more difficult in this case since wehave to deal with a problem that lacks uniform ellipticity, as we willexplain in Section 3.3. Since the G2 form is cubic, the needed quadraticestimate of the 3-dimensional instanton equation appears unavailable inthe Lp setting (see end of Section 4.3). So we set up the problem in theSchauder setting, and in linear theory we need to go further from Lp

estimates to Schauder estimates. This is different from [10].As C ′ should be sufficiently close to C, we assume that they arise in

a one-parameter smooth family of coassociative submanifolds Ct withsmall t. Contracting with the normal vector field n := dCt/dt|t=0 forthe infinitesimal deformation with the G2-form Ω, we obtain a closedself-dual two form ιnΩ ∈ Ω2

+ (C0). Using the induced metric, from n onecan define an almost complex structure J = Jn on C0 away from thezero set of ιnΩ (see (2) for details).

Theorem 1. Suppose that (M,Ω) is a G2-manifold and Ct is aone-parameter smooth family of coassociative submanifolds in M . WhenιnΩ ∈ Ω2

+ (C0) is nonvanishing, then:

1) (Proposition 6) If At is any one-parameter family of associativesubmanifolds (i.e. instantons) in M satisfying

∂At ⊂ Ct ∪ C0, limt→0

At ∩C0 = Σ0 in the C1-topology,

then Σ0 is a Jn-holomorphic curve in C0.2) (Theorem 27) Conversely, every regular Jn-holomorphic curve Σ0

(namely those for which the linearization of ∂Jn on Σ0 is surjec-tive) in C0 is the limit of a family of associative submanifolds At

as described above.

Notice that in the product situation, where

(M,Ω) :=(X × S1,ReΩX + ωX ∧ dθ

)

THIN INSTANTONS AND SEIBERG-WITTEN INVARIANTS 421

(cf. Section 2.2) with (X,ΩX) being a Calabi-Yau threefold, then ourtheorem would follow from the work of [10].

The paper is organized as follows: In Section 2, we first recall somebasics of Floer theory; next we describe their G2-counterparts; then weexplain the connection between instantons and Seiberg-Witten invari-ants; and last we study the deformation of instantons with the aim togeneralize to almost instantons. In Section 3, we first study the lineardifferential operator D (defined in (19)) on a type of thin 3-manifold,which is a linear approximation of the instanton equation; then we givethe L2 and Schauder estimates of its inverse D−1. In Section 4, we firstcompare the linearized instanton equation on almost instantons with theoperator D on linear models, then we use the implicit function theoremto perturb almost instantons to true instantons, thus proving our maintheorem.

Acknowledgments. The first author expresses his gratitude to J.H.Lee, Y.G. Oh, C. Taubes, R. Thomas, and A. Voronov for useful dis-cussions. The second author thanks S.L. Kong, G. Liu, Y.J. Lee, Y.G.Shi, and L. Yin for useful discussions. The third author is grateful forthe nice research environment in math departments of The ChineseUniversity of Hong Kong and the University of Minnesota, and usefuldiscussion with T.J. Li. We thank the anonymous referees for pointingout many inaccuracies in our earlier versions and providing suggestionsfor improvement.

The work of the first author described in this paper was partially sup-ported by grants from the Research Grants Council of the Hong KongSpecial Administrative Region, China (Project No. CUHK401908 andCUHK403709). The work of the second author described in this paperwas partially supported by a grant from the Research Grants Councilof the Hong Kong Special Administrative Region, China (Project No.CUHK403709). The third author was partially supported by an RGCgrant from the Hong Kong Government.

2. Instantons of dimension 2 and 3

2.1. Review of symplectic geometry. Given any symplectic mani-fold (X,ω) of dimension 2n, there exists a compatible metric g so thatthe equation

ω (u, v) = g (Ju, v)

defines a Hermitian almost complex structure

J : TX → TX ,

that is, J2 = −id and g (Ju, Jv) = g (u, v).A holomorphic curve, or instanton of dimension 2, is a two di-

mensional submanifold Σ in X whose tangent bundle is preserved by J .


Equivalently, Σ is calibrated by ω, i.e. ω|Σ = volΣ. By algebraic count-ing of the number of instantons in X, one can define a highly nontrivialinvariant for the symplectic structure on X, called the Gromov-Witteninvariant.

When the instanton Σ has nontrivial boundary, then the correspond-ing boundary value problem would require ∂Σ to lie on a Lagrangiansubmanifold L in X, i.e. dimL = n and ω|L = 0. Floer studied theintersection theory of Lagrangian submanifolds and defined the Floerhomology group HF (L,L′) under certain assumptions.

Suppose that X is a compact Calabi-Yau manifold, i.e. the holonomygroup of the Levi-Civita connection is inside SU (n); equivalently, J isan integrable complex structure on X and there exists a holomorphicvolume form ΩX ∈ Ωn,0 (X) on X satisfying

(−1)n(n−1)

2 (i/2)nΩX ∧ ΩX = ωn/n!.

Under the mirror symmetry transformation, HF (L,L′) is expected tocorrespond to the Dolbeault cohomology group of coherent sheaves inthe mirror Calabi-Yau manifold.

A Lagrangian submanifold L in X is called a special Lagrangiansubmanifold with phase zero (resp. π/2) if ImΩX |L = 0 (resp. ReΩX |L= 0). With suitable choice of orientation of L, L is calibrated by ReΩX |L(resp. ImΩX |L), that is, ReΩX |L is the volume form of L. They playimportant roles in the Strominger-Yau-Zaslow mirror conjecture forCalabi-Yau manifolds [35].

When X is a Calabi-Yau threefold, there are conjectures of Vafa andothers (e.g. [3][14]) that relate the (partially defined) open Gromov-Witten invariant of the number of instantons with Lagrangian bound-ary condition to the large N Chern-Simons invariants of knots in threemanifolds.

2.2. Counting instantons in (almost) G2-manifolds. Notice thata real linear homomorphism J : Rm → R

m being a Hermitian complexstructure on R

m is equivalent to the following conditions: for any vectorv ∈ R

m, we have:

1) Jv is perpendicular to v.2) |Jv| = |v|.We can generalize J to the case involving more than one vector. We

call a skew symmetric bilinear map

× : Rm ⊗ Rm → R

m

a (2-fold) vector cross product if it satisfies:

(i) (u× v) is perpendicular to both u and v.(ii) |u× v| = Area of parallelogram spanned by u and v = |u ∧ v| .


The obvious example of this is the standard vector product on R3. By

identifying R3 with ImH, the imaginary part of the quaternion numbers,we have

u× v = Im vu.

The same formula defines a vector cross product on R7 = ImO, the

imaginary part of the octonion numbers. Brown and Gray [15] showedthat these two are the only possible vector cross product structures onRm up to the automorphism of Rm.Suppose that M is a seven dimensional Riemannian manifold with a

vector cross product × on each of its tangent spaces. The analog of thesymplectic form is a degree three differential form Ω on M defined asfollows:

Ω (u, v, w) = g (u× v,w) .

Definition 2. Suppose that (M,g) is a Riemannian manifold of di-mension seven with a vector cross product × on its tangent bundle.Then

1) M is called an almost G2-manifold if dΩ = 0.2) M is called a G2-manifold if ∇Ω = 0 with ∇ being the Levi-

Civita connection.

Remark 3. M is a G2-manifold if and only if its holonomy groupis inside the exceptional Lie group G2 = Aut (O). The geometry of G2-manifolds can be interpreted as the symplectic geometry on its knotspace (see e.g. [25], [29]).

A typical family of examples of G2-manifolds can be obtained viathe product manifold M := X × S1 with (X,ωX) being a Calabi-Yauthreefold with a holomorphic volume form ΩX , and the G2-form is givenby

Ω = ReΩX + ωX ∧ dθ.Next we define the analogs of holomorphic curves and Lagrangian

submanifolds in the G2 setting.

Definition 4. Suppose that A is a three dimensional submanifoldof an almost G2-manifold M . We call A an instanton or associativesubmanifold if TA is preserved by the vector cross product ×.

Harvey and Lawson [18] showed that A ⊂ M is an instanton if andonly if A is calibrated by Ω, i.e. Ω|A = volA. This is in turn equivalentto τ |TA = 0 in our Lemma 7 for τ defined in (3) (i.e. Corollary 1.7 inSection IV.1.A. of [18] and Corollary 14 in [25]).

In M-theory, associative submanifolds are also called M2-branes. Inthe case whenM = X×S1 with X a Calabi-Yau threefold, Σ×S1 (resp.L×p) is an instanton inM if and only if Σ (resp. L) is a holomorphiccurve (resp. special Lagrangian submanifold with zero phase) in X.


A natural interesting question is to count the number of instantonsin M . In the special case of M = X × S1, these numbers are related tothe conjectural invariants proposed by Joyce [22] by counting specialLagrangian submanifolds in Calabi-Yau threefolds, and the product ofholomorphic curves with S1. This problem has been discussed by manyphysicists. For example, Harvey and Moore discussed in [19] the mirrorsymmetry aspects of these invariants; Aganagic and Vafa in [3] relatedthese invariants to the open Gromov-Witten invariants for local Calabi-Yau threefolds; Beasley and Witten argued in [4] that when there is amoduli of instantons, then one should count them using the Euler char-acteristic of the moduli space. The compactness issues of the moduli ofinstantons are a very challenging problem because the bubbling-off phe-nomena of (3-dimensional) instantons have not been well understood.This makes it very difficult to define an honest invariant by countinginstantons.

Analogous to the Floer homology for Lagrangian intersections, whenan instanton A has a nontrivial boundary, ∂A 6= φ, one should requireit to lie inside a brane or a coassociative submanifold to make it awell-posed elliptic problem (see [17] for Fredholmness and index com-putation), i.e. submanifolds of maximal dimension in M where the re-striction of Ω is zero. Branes are the analog of Lagrangian submanifoldsin symplectic geometry.

Definition 5. Suppose that C is a submanifold of an almost G2-manifold M . We call C a coassociative submanifold if

Ω|C = 0 and dimC = 4.

For example, when M = X × S1 with X a Calabi-Yau threefold,H×S1 (resp. C×p) is a coassociative submanifold inM if and only ifH (resp. C) is a special Lagrangian submanifold with phase π/2 (resp.complex surface) in X. In [25], J.H. Lee and the first author showed

that the isotropic knot space KS1X of X admits a natural holomorphic

symplectic structure. Moreover, KS1H (resp. KS1C) is a complex La-

grangian submanifold in KS1X with respect to the complex structure J(resp. K).

Constructing special Lagrangian submanifolds with zero phase in Xwith boundaries lying on H (resp. C) corresponds to the Dirichlet (resp.Neumann) boundary value problem for minimizing volume among La-grangian submanifolds as studied by Schoen, Wolfson ([33], [34]), andButcher [5]. For a general G2-manifold M , the natural boundary valuefor an instanton is a coassociative submanifold. Similar to the inter-section theory of Lagrangian submanifolds in symplectic manifolds, wepropose to study the following problem: Count the number of instantonsin G2-manifolds bounding two coassociative submanifolds.


The product of a coassociative submanifold with a two dimensionalplane inside the eleven dimension spacetime M × R

3,1 is called a D5-brane in M-theory. Counting the number of M2-branes between twoD5-branes has also been studied in the physics literature.

In general, this is a very difficult problem. For instance, countingS1-invariant instantons in M = X × S1 is the open Gromov-Witteninvariant. However, when the two coassociative submanifolds C and C ′

are close to each other, we can relate the number of thin instantonsbetween them to the number of J-holomorphic curves in C (Theorem1), and hence, by Taubes’ work, to the Seiberg-Witten invariant of C.

2.3. Relationships to Seiberg-Witten invariants. To determinethe number of instantons between nearby coassociative submanifolds,we first recall the deformation theory of compact coassociative subman-ifolds C inside any G2-manifoldM , as developed by McLean [28]. Givenany normal vector field n ∈ Γ

(NC/M

), the interior product ιnΩ is natu-

rally a self-dual two form on C because of Ω|C = 0. This gives a naturalidentification,

Γ(NC/M

) ≃→ Λ2+ (C)

n→ η0 = ιnΩ.(1)

Furthermore, infinitesimal deformations of coassociative submanifoldsare parameterized by self-dual harmonic two forms η0 ∈ H2

+ (C), andthey are always unobstructed, i.e. any such forms with sufficiently smallnorm give actual deformations to nearby coassociative submanifolds (seesection 4 of [28]). Notice that the zero set of η0 is the intersection of Cwith a infinitesimally nearby coassociative submanifold Ct, that is,

η0 = 0 = limt→0

(C ∩ Ct) ,

where C = C0 and η0 = dCt/dt|t=0.Since

η0 ∧ η0 = η0 ∧ ∗η0 = |η0|2 ∗ 1,η0 defines a natural symplectic structure on Creg := C\ η0 = 0. If wenormalize η0 to η,

η = η0/ |η0| ,then the equation

η (u, v) = g (Ju, v)

defines a Hermitian almost complex structure J on Creg. The J is de-termined by η, which in turn is determined by n, so we denote it by Jn.More explicitly, for u ∈ TCreg,

(2) Jn (u) = |n|−1 n× u.


The next proposition says that when two coassociative submanifoldsC and C ′ come together, not necessarily disjoint, then the limit of in-stantons bounding them will be a Jn-holomorphic curve Σ in Creg withboundary C ∩ C ′.

Proposition 6. Let M be a G2-manifold. Suppose that for someε0 > 0, there is a smooth map

ψ : C × [0, ε0] −→M

such that for each t ∈ [0, ε0] , ψt (·) := ψ (·, t) is a smooth immersionof C into M as a coassociative submanifold Ct := ψ (C × t). Supposethat n = dCt/dt|t=0 ∈ Γ

(NC/M

)is nowhere vanishing, and

φt : Σ× [0, t] −→M

is a smooth family of instantons in M such that for each t ∈ (0, ε0], φtis an associative immersion with boundary condition

φt (Σ× 0) ⊂ C0 := ψ (C × 0) , φt (Σ× t) ⊂ Ct := ψ (C × t) .Suppose that the C1-limit of φt (Σ× 0) exists as t → 0. Then Σ0 :=limt→0 φt (Σ× 0) is a Jn-holomorphic curve in C0.

Proof. Let us denote the boundary component of At =Image(φt) inC0 by Σt, i.e. Σt := φt (Σ× 0). Let wt be a unit normal vector fieldfor Σt in At. We claim that wt is perpendicular to C0. To see this, notethat TAt being preserved by the vector cross product implies that

wt = u× v

for some tangent vectors u and v in Σt. In fact, for any unit vectoru ∈ TΣt, v := wt×u ∈ TΣt by associativity condition of At and v ⊥ wt,and

u× v = −u× (u× wt)

= τ (u, u,wt) + g (u,wt) u+ g (u, u)wt

= 0 + 0 + wt = wt,

where we have used in the second line the definition of τ in (3) and τis a (vector-valued) form. Therefore, given any tangent vector w alongC0, we have

g (wt, w) = g (u× v,w) = Ω (u, v, w) = 0,

where the last equality follows from C0 being coassociative and Σt ⊂ C0.Reparameterize φt : Σ × [0, ε] → M when necessary, then for small twe can assume that d

dsφt (z, s)∣∣s=0

is parallel to wt (z) for any z ∈ Σ.Noting that

φt (z, 0) ⊂ C0 and φt (z, t) ⊂ Ct,


we see limt→0 wt (z) is parallel to(dCt

dt

∣∣∣∣t=0

)∣∣∣∣Σ0

= n ∈ Γ(Σ0, NC0/M

).

Therefore, along Σ0,

limt→0

wt (z) = |n (z)|−1 n (z) .

For any ut ∈ TΣt, wt × ut ⊥ wt in associative At, so

wt × ut ∈ TΣt.

Since Σt → Σ0 in C1 topology, for any u ∈ TΣ0, u can be realized asthe limit of ut. Therefore

Jn (u) = |n|−1 n× u = limt→0

(wt × ut) ∈ limt→0

TΣt = TΣ0,

i.e. Σ0 is a Jn-holomorphic curve in C0 with respect to the almost com-plex structure Jn defined in (2). q.e.d.

The reverse of the above proposition is also true (Theorem 27). TheLagrangian analog of it was proven by Fukaya and Oh in [10]. On theother hand, by the celebrated work of Taubes, we expect that the num-ber (counted with algebraic weights) of such holomorphic curves in C0

equals the Seiberg-Witten invariant of C0. We conjecture the followingstatement.

Conjecture: Suppose that C and C ′ are nearby coassociative sub-manifolds in a G2-manifold M . Then the number of instantons countedwith algebraic weights inM with small volume and with boundary lyingon C ∪ C ′ is given by the Seiberg-Witten invariants of C.

The main result of our paper is to solve a special case of the aboveconjecture; namely, we will concentrate on the case that C and C ′ areboth compact and they do NOT intersect. (In the remainder of thepaper, we will always assume C and C ′ are compact and they do NOTintersect.) The basic ideas are (i) the limit of such instantons is a J-holomorphic curve for almost complex structure J compatible to the(degenerated) symplectic form η on C coming from its deformationsas coassociative submanifolds and this process can be reversed; (ii) thenumber of J-holomorphic curves in the four manifold C should be re-lated to the Seiberg-Witten invariant of C by the work of Taubes ([37],[38]). Note that one only gets one symplectic form η (and hence onealmost complex structure J) from a given coassociative deformation ofC, though of course one can get more (from different coassociative de-formations).

Suppose that η is a self-dual two form on C with constant length√2;

in particular, it is a (non-degenerate) symplectic form, and Σ is a smoothholomorphic curve in C, possibly disconnected. If Σ is regular in thesense that the linearized Cauchy-Riemann operator DΣ∂J has trivial


cokernel [36], then Taubes showed that the perturbed Seiberg-Wittenequations,

F+a = q (ψ)− r

√−1η,

DA(a)ψ = 0,

have solutions for all sufficiently large r. Here a is a connection onthe complex line bundle E over C whose first Chern class equals thePoincare dual of Σ, PD [Σ], Fa is the curvature 2-form of E and F+

a isthe projection of Fa to ∧2

+ (C), ψ is a section of the twisted spinor bundle

S+ = E⊕(K−1 ⊗ E

)and DA(a) is the twisted Dirac operator, and q (·)

is a certain canonical quadratic map from S+ to i · ∧2+ (C). The number

of such solutions (counted with algebraic weights) is the Seiberg-Witteninvariant SWC (Σ) of C. Furthermore, the converse is also true; namely,the Seiberg-Witten invariant SWC (Σ) is equal to the Gromov-Witteninvariants counting holomorphic curves Σ. Thus Taubes established anequivalence between Seiberg-Witten theory and Gromov-Witten theoryfor symplectic four manifolds. This result has far reaching applicationsin four dimensional symplectic geometry.

For a general four manifold C with nonzero b+ (C), using a genericmetric, any self-dual two form η on C defines a degenerate symplecticform on C, i.e. η is a symplectic form on the complement of η = 0,which is a finite union of circles (see [12][21]). Therefore, one mightexpect to have a relationship between the Seiberg-Witten invariantsof C and the number of holomorphic curves with boundaries η = 0 inC. Part of this Taubes’ program has been verified in [37], [38].

Suppose that η is a nowhere vanishing self-dual harmonic two formon a coassociative submanifold C in a G2-manifoldM . For any holomor-phic curve Σ in C, we want to construct an instanton in M bounding Cand C ′, where C ′ is a small deformation of the coassociative submani-fold C along the normal direction given by η. Notice that C and C ′ donot intersect. We will construct such an instanton using a perturbationargument which requires a lower bound on the first eigenvalue for theappropriate elliptic operator. Recall that the deformation of an instan-ton is governed by a twisted Dirac operator. We will reinterpret it as acomplexified version of the Cauchy-Riemann operator in Section 3.1.

2.4. Deformation of instantons. To construct an instanton A in Mfrom a holomorphic curve Σ in C, we need to perturb an almost in-stanton A′ to a honest one using a quantitative version of the implicitfunction theorem. Let us first recall the deformation theory of instan-tons A ([18] and [25]) in a Riemannian manifold (M,g) with a parallel(or closed) r-fold vector cross product

× : ΛrTM → TM .


In our situation, we have r = 2. By taking the wedge product with TM ,we obtain a homomorphism τ ,

τ : Λr+1TM → Λ2TM ∼= Λ2T ∗M ,

where the last isomorphism is induced from the Riemannian metric. Asa matter of fact, the image of τ lies inside the subbundle g

⊥M which is

the orthogonal complement of gM ⊂ so (TM) ∼= Λ2T ∗M , the bundle ofinfinitesimal isometries of TM preserving ×. That is,

τ ∈ Ωr+1(M, g⊥M

).

Lemma 7. ([18], [25]) An r + 1 dimensional submanifold A⊂M isan instanton, i.e. TA is preserved by ×, if and only if

τ |A = 0 ∈ Ωr+1(A, g⊥M

).

This lemma is important in describing deformations of an instanton.Mclean [28] used this to show that the normal bundle to an instantonA is a twisted spinor bundle over A and infinitesimal deformations of Aare parameterized by twisted harmonic spinors.

In our present situation, (M,g) is a G2-manifold. Using the crossproduct, we can identify g

⊥M ⊂ Λ2T ∗M ∼= Λ2TM with the tangent

bundle TM , i.e. for u ∧ v ∈ Λ2TM , we identify it with w ∈ TM thatw = u × v. Then we can also characterize τ ∈ Ω3 (M,TM) by thefollowing formula:

(∗Ω) (u, v, w, z) = g (τ (u, v, w) , z) .

More explicitly,

(3) τ (u, v, w) = −u× (v × w)− g (u, v)w + g (u,w) v.

Therefore A⊂M is an instanton if and only if ∗A (τ |A) = 0 ∈ TM |A.Example. The G2 manifold R

7:

R7 ≃ ImO ≃ ImH⊕H = (x1i+ x2j+ x3k, x4 + x5i+ x6j+ x7k) ,

the standard basis consists of ei =∂

∂xi(i = 1, 2, . . . 7), and the multipli-

cation × for (a, b) , (c, d) ∈ ImH⊕H ≃ ImO is

(4) (a, b)× (c, d) = (ac− d∗b, da+ bc∗)

(Cayley–Dickson construction), where z∗ denotes the conjugate of thequaternion z. The G2 form Ω is

Ω = ω123 − ω167 − ω527 − ω563 − ω154 − ω264 − ω374,


and the form τ is the following ((5.4) in [28]):

τ =(ω256 − ω247 + ω346 − ω357

) ∂

∂x1+(ω156 − ω147 − ω345 + ω367

) ∂

∂x2

+(ω245 − ω267 − ω146 − ω157

) ∂

∂x3+(ω567 − ω127 + ω136 − ω235

) ∂

∂x4

+(ω126 − ω467 + ω137 + ω234

) ∂

∂x5+(ω457 − ω125 − ω134 + ω237

) ∂

∂x6

+(ω124 − ω456 − ω135 − ω236

) ∂

∂x7,

(5)

where ωijk = dxi∧dxj∧dxk. ImH⊕0 is associative (i.e. an instanton),and 0⊕H is coassociative.

As a matter of fact, if A is already close to being an instanton, thenwe only need the normal components of ∗A (τ |A) to vanish.

Proposition 8. There is a positive constant δ such that for any 3-plane A in

(R7,Ω

)with |τ |A| < δ, A is an instanton if and only if

∗A (τ |A) ∈ TA.

Proof. McLean observed (from formula (5.6) in [28]) that if At is afamily of linear subspaces in M ∼= R

7 with A0 an instanton, then

∗At

(dτ |At

dt

)∣∣∣∣t=0

∈ NA0/M ⊂ TM |A0 .

We may assume that A is spanned by e1, e2 and e3 = e3+∑7

a=4 taea forsome small ta’s where ei’s are a standard basis for R7, in particular e1×e2 = e3. This is because the natural action of G2 on the GrassmannianGr (2, 7) is transitive. An easy computation (cf. equation (5.4) in [28])shows that the normal component of ∗ (τ |A) in NA/M is given by

∗ (τ |A)⊥ = −t5 (e4)⊥ + t4 (e5)⊥ + t7 (e6)

⊥ − t6 (e7)⊥ ,

where (·)⊥ denote the orthogonal projection onto NA/M . When ta’s are

all zero, we have (ea)⊥ = ea for 4 ≤ a ≤ 7. In particular, they are

linearly independent when ta’s are small. In that case, ∗ (τ |A)⊥ = 0 willactually imply that ta = 0 for all a, i.e. A is an instanton in M . Hencethe proposition. q.e.d.

This proposition will be needed later when we perturb an almostinstanton to an honest one. We also need to identify the normal bundleNA/M to an instanton A with a twisted spinor bundle over A as following[28]: We denote P to be the SO (4)-frame bundle of NA/M . Using theidentification

a : SO (4) = (Sp (1)× Sp (1)) /± (1, 1) → SO (H) ,

(p, q) · y = pyq, with p, q ∈ Sp (1) and y ∈ H.


Mclean [28] showed that the normal bundle NA/M can be identified asan associated bundle to P for the representation SO (4) → SO (H) givenby (p, q) · y = pyq. The spinor bundle S of A is associated to P for therepresentation s : SO (4) → SO (H) given by (p, q) · y = yq. Let E bethe associated bundle to P for the representation e : SO (4) → SO (H)given by (p, q) · y = py. Then because the 3 representations a, s, and ehave the relation

a = s e,we obtain

NA/M∼= S⊗H E.

An alternative proof ofNA/M∼= S⊗HE using explicit frame identification

is contained in the proof of the next theorem.We re-derive McLean’s theorem on deformation of associative sub-

manifolds A in a G2 manifold M . The original proof (Theorem 5.2 in[28]) is not quite precise: the associative form τ ∈ Ω3 (M,TM) is vector-valued rather than a usual differential form, so the pull back operationand Cartan formula need to be clarified. The key is to define a suitablenotion of pull back for vector-valued forms. Our calculation is flexibleand can be extended to almost associative submanifolds in later sections.Other proofs were given in [2] and [16].

Theorem 9 (McLean). Under the correspondence of normal vectorfields with twisted spinors, the Zariski tangent space to associative sub-manifolds at an associative sub-manifold A is the space of harmonictwisted spinors on A, that is, the kernel of the twisted Dirac operator.

Proof. For any section V of NA/M with C0 norm smaller than theinjectivity radius δ0 of M , we define a nonlinear map

F : Γ(NA/M

)→ Ω3 (A, i∗TM) ,

F (V ) = TV (expV )∗ τ,(6)

where for the embedding expV : A → M , (expV )∗ pulls back thedifferential form part of the tensor τ , and TV : Texpp(tV )M → TpM pulls

back the vector part of the tensor τ by parallel transport along thegeodesic expp (tV ). There is an ambiguity of the form part and vectorpart of tensor τ up to a scalar function-valued matrix transform Θ andΘ−1 respectively, but by the linearity of TV and (expV )∗ on scalarfunction factors, one can easily show the definition of F is independenton such Θ, so F is well-defined. We make F more explicit by using a goodframe. At a point p ∈ A, we pick two orthonormal vectors W1,W2 inTpA, then with respect to the induced connection ∇A on A, we paralleltransport W1,W2,W3 =W1 ×W2 from p to a neighborhood B in Aalong geodesic rays from p. From the construction we see

(7) ∇AWiWj (p) = 0, for 1 ≤ i, j ≤ 3.


Then at any q ∈ B ⊂ A, we have orthonormal basis

W1 (q) ,W2 (q) ,W3 (q) =W1 (q)×W2 (q)spanning TqA. (This uses that W1 (q) ×W2 (q) and the parallel trans-portedW3 (p) are both orthogonal toW1 (q) andW2 (q) in 3 dimensionalA). We further choose a smooth normal unit vector field W4 on B in Mas follows: we choose a W4 ∈ NA/M (p) , then use the parallel transport

of NA/M with respect to the induced connection ∇⊥ in NA/M defined

as ∇⊥ = ⊥∇ from the metric on M , where ⊥ : TM → NA/M is thenatural projection. Thus on B,

W4 ⊥ W1,W2,W1 ×W2 .Using Lemma A.15 in [18] (Cayley–Dickson construction), we canuniquely extend Wα (q)1,2,3,4 to basis Wα (q)α=1,2,...,7 of TqM , suchthat

Wi+3 (q) =Wi (q)×W4 (q)

for i = 1, 2, 3, and the correspondence

(8) TqM ∋Wαi↔ eα ∈ ImO, α = 1, 2, . . . , 7

preserves the inner product · and cross product ×, where eαα=1,2,...,7

is the standard basis of R7 ≃ ImO defined in the previous example. Solocally NA/M is trivialized as B × H, and at any q ∈ B, by the abovebasis Wα we have algebra isomorphism

TqM = TqA⊕NA/M (q)i≃ ImH⊕H ≃ ImO

that smoothly depends on q ∈ B. Hence NA/M is a quaternion valuedbundle over A isomorphic to S ⊗H E. From our construction we alsohave

(9) ∇⊥WiWk (p) = 0 for i = 1, 2, 3 and k = 4, 5, 6, 7,

because ∇⊥WiW4 (p) = 0 by construction of W4, and for k = 5, 6, 7, say

k = 5,

∇⊥WiW5 (p) = ⊥ (∇Wi (W1 ×W4)) (p)

= ⊥ (∇WiW1 ×W4 +W1 ×∇WiW4) (p)

= ∇AWiW1 (p)×W4 +W1 ×∇⊥

WiW4 (p) = 0,(10)

where the second row is because NA/M (p) ×W4 ⊂ TpA (for ImO ≃ImH⊕H, (0,H)× (0, 1) ⊂ (ImH, 0) by (4)) and W1 × TpA ⊂ TpA byassociative condition. We remark that the parallel transport in NA/M

w.r.t ∇⊥ is an isometry, for if ∇⊥TW = 0 for section W in NA/M , then

(11) ∇T 〈W,W 〉 = 2 〈∇TW,W 〉 = 2⟨∇⊥

TW,W⟩= 0.


Next, for each q ∈ B, we parallel transport the frame Wα (q)α=1,2,...,7

along geodesical rays emanating from q in M in NA/M (q) directions upto length δ0. This extends the frame to a tubular neighborhood of A inM . Then ∇VWα (q) = 0 for any V ∈ Γ

(NA/M

). If we write

τ = ωα ⊗Wα (α = 1, 2, . . . , 7)

following Einstein’s summation convention, then

F (V ) (q) = (expV )∗ ωα (q)⊗ TVWα

(expq V

).

We have

F ′ (0)V =d

dt

∣∣∣∣t=0

F (tV )

=d

dt

∣∣∣∣t=0

[(exp tV )∗ ωα ⊗ TtVWα]

= LV ωα ⊗Wα + ωα ⊗∇VWα

= d (iV ωα)⊗Wα + iV dω

α ⊗Wα + ωα ⊗∇VWα.(12)

Since ∇τ = 0 and ∇VWα (q) = 0 by the parallel property of τ and Wα,we have

0 = ∇V τ (q) = ∇V ωα⊗Wα (q)+ω

α⊗∇VWα (q) = ∇V ωα (q)⊗Wα (q) ,

and so ∇V ωα (q) = 0. Since ∇ = d + A and in normal coordinates the

connection 1-form A vanishes at q along the fiber direction of NA/M , wehave iV dω

α (q) = 0. Therefore, by (12), at q we have

(13) F ′ (0)V (q) = d (iV ωα)⊗Wα (q) .

By our choice of Wa, the τ = ωa⊗Wα at q is the standard form (5), andby the parallel property of the cross product × and τ , in the neighbor-hood of q in M the τ is also of standard form as (5), in the sense thatthe coordinate vector ∂

∂ωi is replaced by the frame Wi, and ωijk = dxi∧

dxj ∧ dxk is replaced by W ∗i ∧W ∗

j ∧W ∗k where W ∗

α is the dual vector ofWα. Namely,

τ = (W ∗2 ∧W ∗

5 ∧W ∗6 −W ∗

2 ∧W ∗4 ∧W ∗

7 + · · · )⊗W1 + similar terms.

This is because ∇V (τ (Wi,Wj,Wk)) = 0 for V ∈ Γ(NA/M

), by ∇τ = 0

and the parallel property of Wαα=1,2,...,7 in NA/M directions. There-

fore from (13), similar to the way of deriving (5.6) in [28], and notingdW ∗

i (p) = 0 (i = 1, 2, . . . 7) from (7) and (9), we get

F ′ (0)V (p) = d (iV ωα)⊗Wα (p)(14)

= DV (p)⊗W ∗1 ∧W ∗

2 ∧W ∗3 (p)

= DV (p)⊗ dvolA (p) ,


where for V = V 4W4 + V 5W5 + V 6W6 + V 7W7,

DV (p) = −(V 51 + V 6

2 + V 73

)W4 +

(V 41 + V 6

3 − V 72

)W5

+(V 42 − V 5

3 + V 71

)W6 +

(V 43 + V 5

2 − V 61

)W7

is the twisted Dirac operator (5.2) in [28] and also (17) below, withV ki = dV k (Wi), and dvolA = W ∗

1 ∧W ∗2 ∧W ∗

3 is the induced volumeform on A ⊂ M since Wαα=1,2,3 is the orthonormal basis of TA.Since p ∈ A and the section V are arbitrary, we have

(15) F ′ (0) = D ⊗ dvolA.

Note that both sides of the above identity are independent on the choiceof the frame Wαα=1,2,...,7. If the normal vector field V is induced from

deformation of associative submanifolds At0≤t≤ε0, i.e.

At = expU (t) ·A for U (t) ∈ Γ(NA/M

)and U ′ (0) = V,

then differentiating F (U (t)) = 0 at t = 0 and using (15) we get DV = 0on A, namely V is a harmonic twisted spinor. q.e.d.

Remark 10. 1) The vector field V is only defined on A ⊂M , butalong the geodesic rays from A in the fiber directions of NA/M , onecan extend V to an open neighborhood of A by parallel transport;therefore the Lie derivative LV ω for any 3-form ω on M makessense in this neighborhood. However, the Lie derivative of ω re-stricted on A, namely i∗A (LV ω) for the inclusion iA : A → M , isactually independent of the extension of V , as mentioned in [28].One can see this from the Cartan formula

LV ω = d (iV ω) + iV dω

as follows: On A, the second term iV dω is independent on theextension of V . For the first term d (iV ω), since we restrict it onA ⊂ M , one can directly check that this term only involves thederivatives of the section V and ω in tangent directions of A, forany derivative in normal direction will contribute a covector notin T ∗A and make the corresponding summand in i∗A (LV ω) vanish.So i∗A (LV ω) is independent on the extension of V and ω to M .

2) In the preceding proof, the normal vector field V is used only toensure that expV : A→ M is an embedding and give d (iV ω

α)⊗Wα a twisted Dirac operator interpretation on the normal bundleNA/M . Actually, to write down the linearization of F , one onlyneeds the normal bundle of A in M in differential topology sense(namely at any p ∈ A, NA/M (p)+TpA = TpM , but not necessarilyNA/M (p) ⊥ TpA). If we denote the differential topological normal

bundle by N topA/M , then for section V ∈ Γ

(N top

A/M

)with small C0

norm, expV : A → M is an embedding so we can define thelinearization of F as before.


3) The linearization formula

(16) F ′ (0)V = d (iV ωα)⊗Wα + iV dω

α ⊗Wα + ωα ⊗∇VWα

holds for any section V ∈ Γ(N top

A/M

), and any tangent frame

Wai=1,2,...7 along any submanifold A ⊂ M (not necessarily as-

sociative). The term d (iV ωα) ⊗Wα is the principal symbol term

of the first order linear differential operator F ′ (0) and behavesfunctorially under the diffeomorphism between two manifolds. Toget F ′ (0)V (p) = d (iV ω

α)⊗Wα (p), one only needs to choose theframe Wαα=1,...,7 that is parallel along the curves expp (tv) for

all p ∈ A and v ∈ N topA/M (p).

4) On an almost instanton A, its normal bundle is not closed under ×by TA in general, so the linearized instanton equation F ′ (0)V canNOT be interpreted as a twisted Dirac operator. For this reason,we will seldom use the twisted Dirac operator, but mainly (16) todo computations and estimates of F ′ (0)V , with the aid of goodlocal frames Wai=1,2,...7.

5) At a point p in a general 3-manifold A ⊂ M with volume formdvolA, we can write the linearization

F ′ (0)V (p) = G V (p)⊗dvolA (p) ,

where G is a first order linear differential operator, whose principalsymbol depends on the algebraic relation of the frame Wa underthe products × and · , and the volume form on A. Besides the asso-ciative Wai=1,2,3+coassociative Wai=4,5,6,7 type frame, theremay be some other type of frames leading to a meaningful differ-ential operator G.

By McLean’s theorem, the normal bundle to any instanton A is atwisted spinor bundle S over A corresponding to the representationSO (4) → SO (H) given by (p, q) · y = pyq. Let D be the twisted Diracoperator on A. We want to write it down explicitly in local coordinatesnear p. We let Wα7α=1 be a local orthonormal frame constructed asabove. Suppose

V = V 4W4 + V 5W5 + V 6W6 + V 7W7

is a normal vector field to A and we write the covariant differentiation ofV as ∇ (V ) := V α

i Wα ⊗ ωi withωibeing the co-frame dual to Wi;

then the twisted Dirac operator is

(17) D =W1 ×∇1 +W2 ×∇2 +W3 ×∇3,

where ∇i := ∇⊥Wi

(i = 1, 2, 3). For instance, when the G2 manifold isImO ≃ ImH⊕H and the instanton is ImH, then by viewing V as a H

valued function, V = V 4+ iV 5+ jV 6+kV 7, the twisted Dirac operatoris D = i∇1 + j∇2 + k∇3. Expression (17) can be easily shown to be


independent on the choice of orthonormal basis Wai=1,2,3 of TA; thus

D is globally defined on A. From (17) and (9) we have

DV (p) = (W1 ×∇1 +W2 ×∇2 +W3 ×∇3)(V 4W4 + V 5W5 + V 6W6 + V 7W7

)

= −(V 51 + V 6

2 + V 73

)W4 +

(V 41 + V 6

3 − V 72

)W5

+(V 42 − V 5

3 + V 71

)W6 +

(V 43 + V 5

2 − V 61

)W7.(18)

We remark that away from p the expression of DV may have more 0-thorder terms in general.

3. Dirac Equation on Thin 3-manifolds

3.1. A simplified model. To motivate our analytical estimates forlater sections, let us first consider a simplified model. Suppose thatAε = [0, ε]×Σ is a three manifold and (x1, z) are coordinates on Aε. OnAε we put a warped product metric gAε,h = h (z) dx21 + gΣ, where Σ isa Riemann surface with a background metric gΣ and h (z) > 0 is a C∞

function on Σ. Let e1 be the unit tangent vector field on Aε normal toΣ, namely along the x1-direction and e1 = h−1/2 (z) ∂

∂x1. We introduce

a first order linear differential operator D that

(19) D = e1 · ∇1 + ∂ = e1 · h−1/2 (z)∂

∂x1+ ∂,

where ∇1 = ∇e1 = h−1/2 (z) ∂∂x1

, and ∂ =(∂, ∂

∗)is the Dirac operator

on the Dolbeault complex Ω0,∗C

(L) of Hermitian line bundle L over theRiemann surface Σ with a connection ∇ (cf. Proposition 3.67 of [6] orProposition 1.4.25 of [31], by taking the Kahler manifold to be Σ andthe Hermitian line bundle to be L). Here ∂ acts on the spinor bundleSΣ := S

+ ⊕ S− via the following identification:

(20)Ω0C(L)⊕ Ω0,1

C(L)

(∂,∂∗)−→ Ω0,1

C(L)⊕ Ω0

C(L)

↓ ↓S+ ⊕ S

− ∂−→ S− ⊕ S

+

where on the left S+ and S

− are identified to complex line bundles Land L ⊗ Λ0,1

C(Σ) respectively, ∂ : Ω0

C(L) → Ω0,1

C(L) is the Dolbeault

operator, and ∂∗: Ω0,1

C(L) → Ω0

C(L) is its formal adjoint operator. The

Dolbeault operator ∂ depends on the complex structures j on Σ, J onL, and the connection ∇ of L.

When h (z) is a constant, from the spinor bundle SΣ = S+ ⊕ S

−

over Σ one can construct a spinor bundle S over the odd dimensionalmanifold Aε := [0, ε] × Σ by taking the Cartesian product of SΣ with[0, ε] (see Chapter 22 in [8]). When there is no confusion, we also write


S = S+ ⊕ S

− where the S± are the Cartesian products of the S

± ofSΣ with [0, ε]. [8] also constructs a Dirac operator on the spinor bundleS →Aε right from the Dirac operator ∂ on the spinor bundle SΣ→Σ,and it is D =e1 · ∇1 + ∂ as in (19). For general h (z), our D is a Diractype operator but not necessarily a genuine Dirac operator.

Let’s recall the Clifford multiplication of TAε on S. Since S is theCartesian product of SΣ with [0, ε], it is enough to define the Cliffordmultiplication · of e1 on SΣ. Let σ be the volume element of the Cliffordbundle Cl (Σ), σ2 = 1. Then S

+ and S− are the ±1 eigenbundles of σ

and SΣ = S+⊕S

−. Since the Clifford multiplication · of e1 on SΣ satisfiese21 = −1, the natural choice of the action e1· on SΣ is to let S+ and S

− bethe ±i eigenbundles; namely, for (u, v) ∈ S

+⊕S−, e1 · (u, v) = (iu,−iv).

The connection of S along the x1 direction is trivial.Given z0 ∈ Σ, in its neighborhood in Σ we can choose a complex

coordinate z = x2+ ix3 ∈ C with ∂∂x2

, ∂∂x3

orthonormal at z0. We locally

trivialize the spinor bundle SΣ= S+ ⊕ S

− → Σ as

H = C+ Cj = C2 →C.

We may choose the trivialization such that the Dirac operator(∂, ∂

∗)∣∣∣z0

=(∂z,−∂z

).

We may write the section V of S → Aε as V = (u, v) := u+vj ∈S+⊕S−

with

u = V 4 + iV 5 ∈ S+ ≃ C,

v = V 6 + iV 7 ∈ S+ ≃ C, and vj ∈ S

+j = S−.

With these understood, we may write (19) as

DV =

[i 00 −i

](h−1/2 (z)

∂

∂x1+

[0 −i∂

∗

i∂ 0

])[uv

](21)

at z0=

[i 00 −i

](h−1/2 (z)

∂

∂x1+

[0 i∂zi∂z 0

])[uv

]

=((∇1u+ i∂zv) +

(∇1v + i∂zu

)· j)· i

= (i∇1u− ∂zv) +(−i∇1v+∂zu

)· j,(22)

where at z0 we have the identification

∂ = ∂z =1

2(∇2 + i∇3) , − ∂

∗= ∂z =

1

2(∇2 − i∇3)

with ∇i = ∇ ∂∂xi

for i = 2, 3 and ∇1 = h−1/2 (z) ∂∂x1

. We will also denote

−i∂∗= ∂+ and i∂ = ∂−

respectively. They satisfy ∂+ = (∂−)∗.


This implies that the equation DV = 0 is equivalent to the followingequations, an analog of the Cauchy-Riemann equation,

∂−u+ h−1/2 (z)∂v

∂x1=0,

∂+v + h−1/2 (z)∂u

∂x1=0.(23)

We put the boundary condition

(24) v|∂Aε = 0.

(The more precise formulation will be given in (25).) It is similar to thetotally real boundary condition for J-holomorphic maps. When h (z) isa constant, it is a special case of the theory of boundary value problemsfor Dirac operators developed in [8]. For general h (z), the fact that theboundary value problem is Fredholm follows from our elliptic estimatesin Subsection 3.3. Also see [17] for relevant discussion.

Later in Section 4, we will apply the above D to the case whenh (z) = |n (z)|2, where n (z) = d

dtCt

∣∣t=0

is the normal vector field onC0 coming from the deformation of coassociative submanifolds Ct, andthe Hermitian line bundle L is the normal bundle NΣ0/C0

of the Jn-holomorphic curve Σ0 in a coassociative manifold C0, whose connection∇ is the normal connection from the induced metric of Σ0 ⊂ C0.

3.2. First eigenvalue estimates. In this subsection we will establisha quantitative estimate of the eigenvalue of the linearized operator forthe simplified model Aε = [0, ε]×Σ with warped product metric gAε,h =h (z) dx21 + gΣ, where Σ is a compact Riemann surface, and h (z) is asmooth function on Σ with 1

K ≤ h ≤ K for some constant K > 0. Thevolume form of this metric on Aε is

dvolAε = h12 (z) dvolΣdx1.

We introduce the following function spaces for spinors V = (u, v)over Aε.

Definition 11. Let S be the spinor bundle over (Aε, gAε) and V be asmooth section of S.

1) We define the norm

‖V ‖Lm,p(Aε,S):=

∑

α+β≤m

∫ ε

0

∫

Σ

∣∣∣(∇x1)α (∇Σ)

β V∣∣∣ph

12 (z) dvolΣdx1

1/p

and

‖V ‖Cm(Aε,S):=

∑

α+β≤m

sup∣∣∣(∇x1)

α (∇Σ)β V∣∣∣


where ∇x1 and ∇Σ are the covariant differentiation along x1-direction and Σ-directions (i.e. two tangent directions on Σ) re-spectively with respect to the metric gAε , and the Lp-norm is withrespect to gAε too. By the standard Sobolev embedding theorem,we have an ε independent constant C such that for any smoothsection V,

‖V ‖Lp(Aε,S)≤ C ‖V ‖Lm,q(Aε,S)

, for p ≤ 3q

3−mq

‖V ‖Cm(Aε,S)≤ C ‖V ‖Ll,p(Aε,S)

, for p ≥ 3

l −m

as long as ε is bounded below and above by some universal con-stant. For a later purpose, let us fix ε ∈ [1/2, 3/2] .

2) We define the function spaces

Lm,p (Aε,S) :=V = (u, v) ∈ Γ (Aε,S) | ‖V ‖Lm,p(Aε,S)

< +∞

and Lm,p− (Aε,S) (resp.Lm,p

+ (Aε,S) ) to be the closure (with re-spect to the norm ‖·‖Lm,p(Aε)

) of the subspace of smooth sections

V = (u, v) ∈ Γ (Aε,S) such that v ∈ C∞0 (Aε\∂Aε) (resp. u ∈

C∞0 (Aε\∂Aε)), where C∞

0 (Aε\∂Aε) denotes the space of smoothfunctions with compact support inside Aε\∂Aε. Let us also intro-duce the space

Cm (Aε,S) :=V = (u, v) ∈ Γ (Aε,S) | ‖V ‖Cm(Aε,S)

< +∞,

Cm− (Aε,S) :=

V = (u, v) ∈ Γ (Aε,S) | ‖V ‖Cm(Aε,S)

< +∞, v|∂Aε = 0.

Let D be the operator (19) in our linear model. We will impose bound-ary conditions for sections V that D acts on. It is known (cf. [8] Theorem21.5) that the Dirac operators

(25) D± := D|L1,2±

: L1,2± (Aε,S) → L2 (Aε,S)

give well-posed local elliptic boundary problems and their formal adjointoperators are D∗

± = D∓. This boundary condition restricted on smooth

sections V = (u, v) in L1,2− (Aε,S) means

(26) v|∂Aε = 0, i.e. v (0,Σ) = v (ε,Σ) = 0.

The following theorem compares the first eigenvalue for Dirac opera-tor ∂ on the Riemann surface Σ with the operator D on product threemanifold Aε. The theorem refers to the L2 metric which is defined asfollows: Let U, V be two sections in L2 (Aε,S). We let the inner product〈·, ·〉L2(Aε,S)

be

〈U, V 〉L2(Aε,S):=

∫

[0,ε]

∫

Σ〈U, V 〉 (x1, z) h

12 (z) dvolΣdx1,


where in the integral the 〈·, ·〉 is the inner product in fibers of the spinorbundle S. We let

|V (x1, z)|2 := 〈V (x1, z) , V (x1, z)〉‖V ‖2L2(Aε,S)

:= 〈V, V 〉L2(Aε,S).

Theorem 12. Suppose λ∂− is the first eigenvalue of the Laplacian

∆Σ = ∂+∂− acting on the space L1,2− (Σ,S+) and λ∂+ is the first eigen-

value of the Laplacian ∆Σ = ∂−∂+ acting on the space L1,2+ (Σ,S−).

Let

λD± := inf06=V ∈L1,2

± (Aε,S)

‖DV ‖2L2(Aε,S)

‖V ‖2L2(Aε,S)

.

(Notice the boundary conditions (25) and (26) for V .) Then for the

operator D± : L1,2± (Aε,S) → L2 (Aε,S), we have

λD± ≥ min1

K

λ∂± ,

2

Kε2−K

∥∥∥h−12

∥∥∥2

C1(Σ)

,

where K > 0 is some constant such that 1K ≤ h (z) ≤ K for all z ∈ Σ.

Proof. In the following we will assume the volume form dvolAε =dvolΣdx1 to simplify calculation. General cases can be reduced to thiscase by observing(27)

‖DV ‖2L2(Aε,S)

‖V ‖2L2(Aε,S)

≥∫Aε|DV |2K−1/2dvolΣdx1∫Aε|V |2K1/2dvolΣdx1

=1

K

∫Aε|DV |2 dvolΣdx1∫

Aε|V |2 dvolΣdx1

from the condition 1K ≤ h (z) ≤ K. It is enough to consider the case

D− : L1,2− (Aε,S) → L2 (Aε,S). The D+ case is similar. For any V =

(u, v) ∈ L1,2− (Aε) , we have

〈DV,DV 〉L2(Aε,S)=

∫

Aε

(h−1 (z)

∣∣∣∣∂V

∂x1

∣∣∣∣2

+ 2

⟨h−

12 (z)

∂V

∂x1,

[0 ∂+

∂− 0

]V

⟩

+∣∣∂+v

∣∣2 +∣∣∂−u

∣∣2)dvolAε


Using the formula ∂− = (∂+)∗, we have

∫

Aε

⟨h−

12 (z)

∂V

∂x1,

[0 ∂+

∂− 0

]V

⟩dvolAε

=

∫

Aε

(⟨h−

12∂u

∂x1, ∂+v

⟩+

⟨h−

12∂v

∂x1, ∂−u

⟩)dvolAε

( ∵ S = S+⊕S

− orthogonal decomposition)

=

∫

Aε

(⟨∂−(h−

12∂u

∂x1

), v

⟩−⟨h−

12 v, ∂−

(∂u

∂x1

)⟩)dvolAε

+

∫

ε×Σ

⟨h−

12 v, ∂−u

⟩dvolΣ −

∫

0×Σ

⟨h−

12 v, ∂−u

⟩dvolΣ

=

∫

Aε

(⟨∂−(h−

12

)· ∂u∂x1

, v

⟩+

⟨h−

12∂−

(∂u

∂x1

), v

⟩

−⟨h−

12 v, ∂−

(∂u

∂x1

)⟩)dvolAε (since v vanishes on ∂Aε)

=

∫

Aε

⟨∂−(h−

12

)· ∂u∂x1

, v

⟩dvolAε

( ∵ the above 2nd and 3rd terms cancel).(28)

Therefore

2

∣∣∣∣∫

Aε

⟨h−

12 (z)

∂V

∂x1,

[0 ∂+

∂− 0

]V

⟩dvolAε

∣∣∣∣

≤ 2

∫

Aε

∣∣∣∣⟨∂−(h−

12

)· ∂u∂x1

, v

⟩dvolAε

∣∣∣∣

≤∥∥∥h−

12

∥∥∥C1(Σ)

∫

Aε

∣∣∣∣2⟨∂u

∂x1, v

⟩dvolAε

∣∣∣∣

≤∫

Aε

(1

K|ux1 |2 +K

∥∥∥h−12

∥∥∥2

C1(Σ)|v|2)dvolAε .(29)

In order to estimate∫Aεh−1 (z) |Vx1 |2 dvolAε , we notice that, for any fixed

point p ∈ Σ, v|[0,ε]×p can be treated as a C-valued function over the


interval [0, ε] with boundary value v (0, z) = 0. Hence

∫ ε

0|v|2 dx1 =

∫ ε

0

∣∣∣∣∫ x1

0

∂v

∂x1(t) dt

∣∣∣∣2

dx1

≤∫ ε

0

(∫ x1

0ds

)(∫ x1

0

∣∣∣∣∂v

∂x1(t)

∣∣∣∣2

dt

)dx1

≤∫ ε

0x1dx1

∫ ε

0

∣∣∣∣∂v

∂x1(t)

∣∣∣∣2

dt

≤ Kε2

2

∫ ε

0h−1 (z)

∣∣∣∣∂v

∂x1(t)

∣∣∣∣2

dt,(30)

where the last inequality is by 1K ≤ h (z) ≤ K for all z ∈ Σ. Putting

(29) , (30) in 〈DV,DV 〉L2(Aε,S), we have

〈DV,DV 〉L2(Aε,S)

≥∫

Aε

(h−1 (z) |ux1 |2 +

∣∣∂−u∣∣2 + h−1 (z) |vx1 |2 +

∣∣∂+v∣∣2

− 2

∣∣∣∣⟨h−

12 (z)

∂V

∂x1,

[0 ∂+

∂− 0

]V

⟩∣∣∣∣

)dvolAε

≥∫

Aε

((h−1 (z)− 1

K

)|ux1 |2 +

∣∣∂−u∣∣2 + h−1 (z) |vx1 |2

−K∥∥∥h−

12

∥∥∥2

C1(Σ)|v|2

)dvolAε(by (29))

≥ 0 + λ∂−

∫

Aε

|u|2 dvolAε +(

2

Kε2−K

∥∥∥h−12

∥∥∥2

C1(Σ)

)∫

Aε

|v|2 dvolAε

(by definition of λ∂− and (30))

≥ min

λ∂− ,

2

Kε2−K

∥∥∥h−12

∥∥∥2

C1(Σ)

(∫

Aε

(|u|2 + |v|2

)dvolAε

)

= min

λ∂− ,

2

Kε2−K

∥∥∥h−12

∥∥∥2

C1(Σ)

‖V ‖2L2(Aε,S)

.

For general volume form dvolAε = h12 (z) dvolΣdx1, by (27) there is an

extra factor 1K for the lower bound of the L2 eigenvalue λD± . Hence the

result. q.e.d.

Suppose V = (u, v) ∈ C∞ (Aε,S) ∩ L1,2− (Aε,S) and W = (f, g) ∈

C∞ (Aε,S)∩L2 (Aε,S) and D is the operator (19). Then with respect to


the induced volume form dvolAε = h1/2 (z) dΣdx1 on (Aε, gAε,h), we have∫

Aε

〈DV,W 〉 dvolAε

=

∫ ε

0

∫

Σ

[⟨i(h−1/2ux1

+ ∂+v), f⟩−⟨i(h−1/2vx1

+ ∂−u), g⟩]h1/2dΣdx1

= i

∫ ε

0

∫

Σ

[−〈u, fx1

〉+⟨h1/2v, ∂−f

⟩−⟨h1/2u, ∂+g

⟩+ 〈v, gx1

〉]dΣdx1

+ i

∫

Σ

(〈u, f〉 |ε0 − 〈v, g〉 |ε0) dΣ

= i

∫ ε

0

∫

Σ

[⟨u,−fx1

− h1/2∂+g⟩+⟨v, gx1

+ h1/2∂−f⟩]dΣdx1

+ i

∫

Σ

(〈u, f〉 |ε0 − 〈v, g〉 |ε0) dΣ

=

∫ ε

0

∫

Σ

⟨u, i(h−1/2fx1

+ ∂+g)⟩

−⟨v, i(h−1/2gx1

+ ∂−f)⟩(

h1/2dΣdx1

)

+ i

∫

Σ

(〈u, f〉 |ε0 − 〈v, g〉 |ε0) dΣ

=

∫

Aε

〈V,DW 〉 dvolAε + i

∫

Σ

(〈u, f〉 |ε0 − 〈v, g〉 |ε0) dΣ(31)

since h is independent of x1. This is the Green’s formula.When V ∈ L1,2

− (Aε,S) and W ∈ L1,2+ (Aε,S) , the above boundary

terms are zero, so we have∫

Aε

〈DV,W 〉 dvolAε =

∫

Aε

〈V,DW 〉 dvolAε .

This implies thatD is a self-adjoint operator from L1,2± (Aε,S) to L

1,2∓ (Aε,S)

in the sense of [8]. Since L1,2+ (Aε,S) is dense in L2 (Aε,S) , this implies

thatD : L1,2− (Aε,S) → L2 (Aε,S) is surjective if and only if kerD|L1,2

+ (Aε,S)=

0 ⊂ L1,2+ (Aε,S). By Theorem 12, for small enough ε, we have

kerD|L1,2+ (Aε,S)

= 0 ⇐ ker ∂−∂+ = 0 ⇔ ker ∂+ = 0,

where the last “⇔” is because (∂+)∗= ∂−. Hence we have obtained the

following result.

Theorem 13. Let λ∂− and λ∂+ be the first eigenvalue for ∂+∂− and

∂−∂+ respectively. For D : L1,2− (Aε,S) → L2(Aε,S), if ε is sufficiently

small, then we have

ker ∂− = 0 ⇔ λ∂− > 0 ⇒ D injective,

ker ∂+ = 0 ⇔ λ∂+ > 0 ⇒ D surjective.

Especially if both λ∂− , λ∂+ > 0, then D is one-to-one and onto.


3.3. Schauder estimates for the linear model. In this section wewill develop the necessary linear theory for the equation

DV =W on Aε = Σ× [0, ε]

with a warped product metric gAε,h := h (z) dx21 + gΣ, where D is theoperator (19) in our linear model. The key issue is to estimate the op-erator norm of the inverse operator of D with explicit dependence on ε,as ε goes to zero. When ε is further away from zero, say ε ∈ [1/2, 3/2],we have ε-free Schauder estimates. For ε small, we overcome the dif-ficulty coming from ε by choosing an appropriate integer k so thatkε ∈ [1/2, 3/2] and we extend any solution V = (u, v) on Aε to Akε

in an Lp sense by suitable reflection. However, much care will be neededto obtain the Cα-estimate, because after the reflection of W across theboundary of Aε, it will no longer be continuous in general. This prob-lem will be resolved in the case (ii) part of the proof of the followingtheorem.

Estimating the operator norm of D−1 in the Schauder setting ratherthan in the Lp setting is crucial. Since our goal is to construct instantonsA governed by the equation τ |A = 0, which involves the associative 3-form τ , for deformations of an approximate solution A by normal vectorfields V in M that are in W 1,p class, the nonlinear equation will havecubic terms of ∇V that are outside the Lp space. The cubic terms alsocause difficulty for obtaining desired quadratic estimates for the implicitfunction theorem in the Lp setting (see Remark 25).

The Schauder estimates are harder to obtain than for the Cauchy-Riemann type equations, partly because in our equation (23), the de-rivative of v only controls the ∂x1 and ∂z derivatives of u, not the fullderivatives.

We recall the definition of Holder norms for functions f on a domainΩ in R

n:

[f ]α;Ω := supx,y∈Ωx 6=y

|f (x)− f (y)||x− y|α ,(32)

[f ]Cα(Ω) = ‖f‖C0(Ω) + [f ]α;Ω ,

[f ]C1,α(Ω) = ‖f‖C1(Ω) + [∇f ]α;Ω .Using trivialization of the bundle S over Aε, the Holder norms for sec-tions V of the bundle S are defined by patching the (finitely many)C1,α-coordinate charts on Aε.

Theorem 14. Let D : L1,2− (Aε,S) → L2 (Aε,S) be the operator (23)

defined on Aε = Σ× [0, ε] with warped product metric gAε,h := h (z) dx21+gΣ. Suppose that the first eigenvalues for ∂−∂+ and ∂+∂− are boundedbelow by λ > 0. Then for any 0 < α < 1 and p > 3 there is a positiveconstant C = C (α, p, λ, h) independent of ε such that for any V ∈


C1,α− (Aε,S) and W ∈ Cα (Aε,S) satisfying

DV =W,

we have

C ‖V ‖C1,α− (Aε,S)

≤ ε−(

3p+2α

)

‖W‖Cα(Aε,S).

In other words, there exists a right inverse

Qε : Cα (Aε,S) → C1,α

− (Aε,S)

of D : C1,α− (Aε,S) → Cα (Aε,S) such that ‖Qε‖ ≤ Cε

−(

3p+2α

)

.

Proof. For the operator (23), it is known (cf. [8] Theorem 21.5) thatthe Dirac operators

D± := D|L1,2±

: L1,2± (Aε,S) → L2 (Aε,S)

give well-posed local elliptic boundary problems. Using the orthogonaldecomposition S = S

+⊕S−, we write sections V = (u, v) ∈ C∞

− (Aε,S)andW = (w1, w2) ∈ C∞ (Aε,S), where u,w1 are sections of S

+ and v,w2

are sections of S−. So the equation DV =W may be explicitly writtenas (cf. equation (23))

h−1/2 (z)ux1 + ∂+v = w1

h−1/2 (z) vx1 + ∂−u = w2with v|∂Aε = 0.

To make the exposition more transparent, we will assume that h ≡ 1,and it is clear from the proof below that the argument works equallywell for any h (z) ∈ C∞ (Σ) such that 1

K ≤ h (z) ≤ K for some constantK > 0. For any 0 < ε < 3/2, we take an integer k = k (ε) > 0 (whichdepends on ε) such that

k (ε) ε ∈ [1/2, 3/2] .

For notation brevity, we will write k for k (ε) in the remainder of our pa-per. Then Akε = Σ× [0, kε] is a product region whose second componenthas length uniformly bounded below and above for any ε. We divide theestimates into two cases: Case (i): Suppose that w1 = 0. Then we willhave along the boundary ∂Aε, ux1 = 0 since v = 0. Since v|∂Aε = 0, wecan extend v from Aε to Akε by odd reflection along the walls Σ × jεwith 0 ≤ j ≤ k − 1, while still keeping v in C1,α (Akε). Similarly, weconsider an even extension of u to Akε; then u is still in C1,α (Akε) sinceux1 |∂Aε = 0. The extension formula is

v (x, z) =

−v ((2j + 2) ε− x, z) for x ∈ [(2j + 1) ε, (2j + 2) ε]

v (x− 2jε, z) for x ∈ [2jε, (2j + 1) ε],

u (x, z) =

u ((2j + 2) ε− x, z) for x ∈ [(2j + 1) ε, (2j + 2) ε]

u (x− 2jε, z) for x ∈ [2jε, (2j + 1) ε].

This will induce an even extension of w2 so that the equation DV =Wis satisfied in the Cα sense on Akε. The motivation of even and odd


extension is to have an ε-independent Lp and Schauder estimate. Bydifferentiating both sides of the equation vx1 + ∂−u = w2 with respectto x1, we obtain an equation which is equivalent to the Dirichlet problemof the second order elliptic equation

vx1x1 − ∂−∂+v =∂w2

∂x1and v|∂Akε = 0,

ux1x1 − ∂+∂−u = −∂+w2,

noting that ∂−∂+ is a positive operator. (Since u, v are only in C1,α,they should be understood as weak solutions of the above equations.)Since the Cα-norm is preserved under the above extension, Schauderand Lp interior estimate for the second order elliptic equation on theregion

Akε ⊂ A2kε ∪ A−2kε

would then imply that there are constants C (α) and C (p) independentof ε (because kε ∈ [1/2, 3/2]) such that

‖w2‖Cα(Aε,S−) + ‖V ‖C0(Aε,S)= ‖w2‖Cα(A2kε∪A−2kε,S−) + ‖V ‖C0(A2kε∪A−2kε,S)

≥ C (α) ‖V ‖C1,α(Akε,S)

= C (α) ‖V ‖C1,α− (Aε,S)

,(33)

and

‖w2‖Lp(A2kε∪A−2kε,S−) + ‖V ‖Lp(A2kε∪A−2kε,S)≥ 4C (p) ‖V ‖

L1,p− (Akε,S)

(cf. [13] section 8.11 Theorem 8.32 and [30] Theorem B.3.2 respec-tively), so

‖w2‖Lp(Aε,S−) + ‖V ‖Lp(Aε,S)≥ C (p) ‖V ‖L1,p

− (Aε,S)

by periodicity of reflection. We also have

‖V ‖C0(Aε,S)≤ ‖V ‖

C1−3/p− (Aε,S)

≤ C (p, λ) ‖W‖C0(Aε,S),

whose proof is identical to that of (38) in the following case (ii). Pluggingthis into (33), we have

(34) C (α) ‖V ‖C1,α− (Aε,S)

≤ (C (p, λ) + 1) ‖w2‖Cα(Aε,S−) .

Case (ii): Suppose that w2 = 0 and w1 ∈ Cα (Aε,S+) . Since v = 0, this

implies that if we consider the odd extension of v and the even extensionof u to A2kε∪ A−2kε as in the previous case, then they induce an oddextension of w1 so that the equation

(35)

ux1 + ∂+v = w1

vx1 + ∂−u = 0

is satisfied in the weak sense on A2kε ∪ A−2kε. Notice that w1 does notvanish on ∂Aε in general, so after the odd extension, w1 is no longercontinuous, but still we have w1 ∈ Lp (A2kε ∪ A−2kε,S) for ∀p. Since Akε


is a proper subdomain in A2kε ∪ A−2kε and kε ∈ [1/2, 3/2] , the interior

Lp-estimate then implies that there is a constant C (p) independent ofε such that

‖w1‖Lp(A2kε∪A−2kε,S−) + ‖V ‖Lp(A2kε∪A−2kε,S)≥ 4C (p) ‖V ‖

L1,p− (Akε,S)

.

By the periodicity of w1 and V, that is

(36) ‖w1‖Lp(Akε,S−) + ‖V ‖Lp(Akε,S)≥ C (p) ‖V ‖L1,p

− (Akε,S).

Then we proceed to the Lp estimates of V purely in terms of W . Itfollows from (36) and Theorem 12 that for ε < 3/2, we have

‖W‖L2(Aε,S)≥ C (λ) ‖V ‖L2(Aε,S)

for V ∈ C∞− (Aε,S)

with the constant C (λ) dependent on λ∂+ but independent of ε. To gofrom L2 to Lp, note the following interpolation inequality:

‖V ‖Lp(Akε,S)=

(∫

Akε

|V |p)1/p

≤ ‖V ‖p−1p

C0(Akε,S)·(∫

Akε

|V |)1/p

≤ p− 1

pδ

pp−1 ‖V ‖

p−1p

· pp−1

C0(Akε,S)+

1

p

1

δp

(∫

Akε

|V |) p

p

(by Young’s Inequality)

=p− 1

pδ

pp−1 ‖V ‖C0(Akε,S)

+1

p

1

δp‖V ‖L1(Akε,S)

≤ C(δ

pp−1 ‖V ‖L1,p(Akε,S)

+ δ−p ‖V ‖L2(Akε,S)

)

(by Sobolev embedding),

where C is independent of ε in the last inequality because kε ∈ [1/2, 3/2].Putting this into (36), we have(C (p)− Cδ

pp−1

)‖V ‖L1,p(Akε,S)

≤ ‖W‖Lp(Akε,S)+ Cδ−p ‖V ‖L2(Akε,S)

≤ ‖W‖Lp(Akε,S)+Cδ−p

C (λ)‖W‖L2(Akε,S)

≤ ‖W‖Lp(Akε,S)

(1 +

Cδ−p

C (λ)

).

So

(37) ‖V ‖L1,p− (Akε,S)

≤(C (p)− Cδ

pp−1

)−1(1 +

Cδ−p

C (λ)

)‖W‖Lp(Akε,S)

.

Fix a small δ such that C (p)− Cδp

p−1 > 0 and let the constant

C (p, λ) :=(C (p)− Cδ

pp−1

)−1(1 +

Cδ−p

C (λ)

)


which is independent of ε. Then for p > 3, by Sobolev embedding

C1−3/p− (Akε,S) → L1,p

− (Akε,S) and (37), we have

‖V ‖C

1−3/p− (Aε,S)

= ‖V ‖C

1−3/p− (Akε,S)

≤ C ‖V ‖L1,p− (Akε,S)

≤ C · C (p, λ) ‖W‖Lp(Akε,S)

≤ C (p, λ) ‖W‖C0(Akε,S)= C (p, λ) ‖W‖C0(Aε,S)

,(38)

where the constant C (p, λ) = C · C (p, λ)(32vol (Σ)

) 1p is independent of

ε. We remark that the above argument also works in case (i), so (38)holds in all cases. In the following, we give the Schauder estimate ofV = (u, v). Differentiating the second equation in (35) with respect tox1, we get the Dirichlet problem of the second order elliptic equationfor (weak solutions) v and u:

vx1x1 − ∂−∂+v = −∂−w1 and v|∂Aε = 0,(39)

ux1x1 − ∂+∂−u = ∂x1w1.

Recall that in [13] chapter 4, the non-dimensional Schauder norm‖f‖′

Ck,α(Ω) is defined as

(40) ‖f‖′Ck,α(Ω) = Σk

j=0dj∣∣Dju

∣∣0;Ω

+ dk+α[Dku

]α;Ω

,

where d is the diameter of Ω. For component v, since v|∂Aε = 0, thestandard Schauder estimate for second order elliptic equation on radiusε half balls Bε with centers on ∂Aε (cf. [13] Section 8.11 for regularityof weak solutions and Section 4.4 equation (4.43)) gives

ε ‖w1‖′Cα(Bε,S+) + ‖v‖C0−(Bε,S)

≥ C (α) ‖v‖′C1,α

− (Bε/2,S),

or equivalently, by the definition of ‖f‖′Ck,α(Ω),

ε ‖w1‖Cα(Bε,S+)′ + ‖v‖C0−(Bε,S)

≥ C (α)[‖v‖C0

−(Bε/2,S) + ε ‖∇v‖C0−(Bε/2,S) + ε1+α [∇v]α;(Bε/2,S)

],

(41)

and hence

(42) ε ‖w1‖Cα(Bε,S+) + ‖v‖C0−(Bε,S)

≥ C (α) ε1+α ‖v‖C1,α− (Bε/2,S)

,

where the constant C (α) is independent of ε. Covering Aε by suchradius-ε half balls and radius-ε/2 full balls centered on meridian Σ ×ε/2 and taking supremum on Aε, we have

(43) ε ‖w1‖Cα(Aε,S+) + ‖v‖C0−(Aε,S)

≥ C (α) ε1+α ‖v‖C1,α− (Aε,S)

.


By combining this with the fact that v|∂Aε=0, and the definition of theSchauder norm, we have

(44) ‖v‖C0−(Aε,S)

≤ ‖v‖C

1−3/p− (Aε,S)

ε1−3/p ≤ C (p, λ) ‖w1‖C0(Aε,S)ε1−

3p .

Plugging these into (43), we obtain(45)

C (α) ‖v‖C1,α

− (Aε,S)≤ ε−α ‖w1‖Cα(Aε,S+) + C (p, λ) ‖w1‖C0(Aε,S)

ε−(

3p+α

)

.

For the component u, its boundary value is nonzero, so we cannot di-rectly apply the above inequalities as v. First we derive the C0 estimateof u, using the assumption that ∂+∂− has trivial kernel on Σ. Considerthe section u of the bundle S

+ → Σ defined as

u (z) =

∫ ε

0u (x1, z) dx1.

From equation (35), we have

∂−u (z) = −∫ ε

0h−1/2 (z) ∂x1v (x1, z) dx1

= h−1/2 (z) (v (0, z)− v (ε, z)) = 0.

But from the assumption that ∂+∂− has trivial kernel on Σ, we get that

u (z) ≡ 0.

Let Re u and Imu be the real and imaginary parts of the section u(notice that S+ is a complex line bundle). Then for any fixed z ∈ Σ,

∫ ε

0Re u (x1,z) dx1 = 0 =

∫ ε

0Imu (x1,z) dx1.

By the mean value theorem for the R-valued function Re u (x1,z), thereexists s ∈ [0, ε] depending on z, such that Re u (s, z) = 0. Therefore, forx1 6= s, by the definition of the Schauder norm, we have

|Reu (x1, z)| =|Reu (x1, z)− Re u (s, z)|

|x1 − s|1−3p

· |x1 − s|1−3p

≤ ‖Reu‖C1−3/p(Aε,S)ε1− 3

p .

Then we have

‖Reu‖C0(Aε,S)≤ ‖Re u‖C1−3/p(Aε,S)

ε1−3p

≤ ‖V ‖C

1−3/p− (Aε,S)

ε1− 3

p

≤ C (p, λ) ‖w1‖C0(Aε,S)ε1−

3p

by (38). Similarly,

‖Imu‖C0−(Aε,S+) ≤ C (p, λ) ‖w1‖C0(Aε,S)

ε1− 3

p .


Combining these, we get

(46) ‖u‖C0−(Aε,S+) ≤ C (p, λ) ‖w1‖C0(Aε,S)

ε1−3p .

Now we are ready to derive the Schauder estimate of u. Using the equa-tion

ux1 + ∂+v = w1

vx1 + ∂−u = 0with v|∂Aε = 0,

the Schauder estimate of ux1 and ∂−u can be reduced to that of v. Forthe full covariant derivative ∇Σu on Σ, we observe that

[∇Σu]α;(Aε,S) ≤ [∇Σu]zα;(Aε,S)

+ [∇Σu]x1

α;(Aε,S)

where [·]zα;(Aε,S) and [·]x1

α;(Aε,S)as defined in (32) are the Schauder α-

components for the Σ and [0, ε] directions, respectively. For [∇Σu]zα;(Aε,S)

,

on each slice Σs := Σ × s, we use the elliptic estimate on compactclosed surface Σs to control ∇Σu by ∂−u,

[∇Σu]zα;(Σs,S)

≤ |u|C1,α(Σs,S)≤ C

(∣∣∂−u∣∣Cα(Σs,S)

+ |u|C0(Σs,S)

),

and then take the sup for 0 ≤ s ≤ ε to get

[∇Σu]zα;(Aε,S)

≤ C(∣∣∂−u

∣∣Cα(Aε,S)

+ |u|C0(Aε,S)

)

= C(|vx1 |Cα(Aε,S)

+ |u|C0(Aε,S)

).(47)

Since the estimate for vx1 is known in (45), the only term left to estimateis [∇Σu]

x1

α;(Aε,S). For this we reduce u to the zero boundary value case by

introducing

u = u− ρ(x1ε

)u (ε, z) −

(1− ρ

(x1ε

))u (0, z) ,

where ρ : [0, 1] → [0, 1] , ρ (0) = 0, ρ (1) = 1 is a smooth cut-off functionsuch that ‖ρ‖C2,α[0,1] ≤ C . Then u satisfies

ux1x1 − ∂+∂−u = ∂x1w1 + g and u|∂Aε = 0,

where

w1 = w1 + ρ(x1ε

)[(ux1 (ε, z) − ux1 (0, z)) + (w1 (0, z) − w1 (ε, z))]

and

g =1

ε2ρ′′(x1ε

)[u (0, z) − u (ε, z)]

+1

ερ′(x1ε

)[(ux1 (0, z)− ux1 (ε, z)) + (w1 (ε, z)− w1 (0, z))] .

In the above derivation of w1 and g, we have used the equation

∂+∂−u = ux1x1 − ∂x1w1


from (39). By [13] section 4.4 equation (4.46), we have, on radius-ε halfball Bε,

‖u‖C0−(Bε,S)

+ ε ‖w1‖′Cα(Bε,S+) + ε2 |g|C0−(Bε,S)

≥ C (α) ‖u‖′C1,α

− (Bε/2,S).

By the definition of the norm ‖f‖′Ck,α(Ω) in (40), this is equivalent to

‖u‖C0−(Bε,S)

+ ε ‖w1‖C0(Bε,S+) + ε1+α [w1]α;(Bε,S+) + ε2 |g|C0−(Bε,S)

≥ C (α)[‖u‖C0

−(Bε/2,S) + ε ‖∇u‖C0−(Bε/2,S) + ε1+α [∇u]α;(Bε/2,S)

].

(48)

Therefore

‖u‖C0−(Bε,S)

+ ε ‖w1‖C0(Bε,S+) + ε1+α [∇w1]α;(Bε,S+) + ε2 |g|C0−(Bε,S)

≥ C (α) ε1+α [∇Σu]α;(Bε/2,S) .

(49)

We are going to get rid of the tilde terms in the above inequality by thefollowing rule: The terms with tilde differ from the original terms by w1,u, and their derivatives. If the derivative is with respect to x1 or ∂−,then we reduce it to the previous estimates of v using (35). If it is withrespect to ∇Σ, the full derivative on Σ, then we use the elliptic estimateon each slice of Riemann surface t × Σ. Notice that all ε powers arecoupled with the order of derivatives so all terms are dimensionless,and after the enlargements the estimate of u is of correct ε power. Thefollowing is the precise calculation: From the definition of g, w1, and uwe can easily check that∥∥ε2g

∥∥C0(Bε,S)

≤ C(‖u‖C0(Bε,S)

+ ε ‖ux1‖C0(Bε,S)+ ε ‖w1‖C0(Bε,S)

)

|w1|C0(Bε,S)≤ 3 ‖w1‖C0(Bε,S)

+ ‖ux1‖C0(Bε,S)

[w1]x1

α;(Bε,S)≤ C

([w1]

x1

α;(Bε,S)+

1

εα‖ux1‖C0(Bε,S)

+1

εα‖w1‖C0(Bε,S)

)

[w1]zα;(Bε,S)

≤ C([w1]

zα;(Bε,S)

+ [ux1 ]zα;(Bε,S)

)

‖u‖C0(Bε,S)≤ 2 ‖u‖C0(Bε,S)

‖∇Σu‖C0(Bε,S)≤ 2 ‖∇Σu‖C0(Bε,S)

,

‖∇x1 u‖C0(Bε,S)≤ C

(‖∇x1u‖C0(Bε,S)

+1

ε‖u‖C0(Bε,S)

)

[∇Σu]x1

α;(Bε/2,S)≤ C

([∇Σu]

x1

α;(Bε/2,S)+

1

εα‖∇Σu‖C0(Bε/2,S)

).

(50)

Hence, putting these back into (49) and (48), and noticing that from(41) we can control ‖ux1‖C0(Bε,S)

and ‖∇Σu‖C0(Bε,S)that appeared in


the above inequalities by the following:

ε ‖ux1‖C0(Bε,S)≤ C

(ε∥∥∂+v

∥∥C0(Bε,S)

+ ε ‖w1‖C0(Bε,S)

)(by (35) )

(by (41) ) ≤ C[(

‖v‖C0(Aε,S)+ ε ‖w1‖Cα(Aε,S)

)+ ε ‖w1‖C0(Bε,S)

]

≤ C(‖v‖C0(Aε,S)

+ ε ‖w1‖Cα(Aε,S)

),(51)

and

ε ‖∇Σu‖C0(Σs,S)≤ ε ‖∇Σu‖Cα(Σs,S)

≤ Cε(∥∥∂−u

∥∥Cα(Σs,S)

+ ‖u‖C0(Σs,S)

)(by ellipticity on Σs)

(by (35)) = C(ε ‖vx1‖Cα(Σs,S)

+ ε ‖u‖C0(Σs,S)

)

(by (42)) ≤ C(ε−α ‖v‖C0(Aε,S)

+ ε1−α ‖w1‖Cα(Aε,S)+ ε ‖u‖C0(Σs,S)

),

(52)

then from (49) and (50), we get the estimate for [∇Σu]x1

α;(Bε/2,S):

ε1−α ‖w1‖Cα(Aε,S+) + ε ‖u‖C0−(Aε,S)

+ ε−α ‖v‖C0−(Aε,S)

≥ CC (α) ε1+α [∇Σu]x1

α;(Bε/2,S).

Combining (47) about [∇Σu]zα;(Bε/2,S)

, we have the full control of

[∇Σu]α;(Bε/2,S):

ε1−α ‖w1‖Cα(Aε,S+) + ε ‖u‖C0−(Aε,S)

+ ε−α ‖v‖C0−(Aε,S)

≥ CC (α) ε1+α [∇Σu]α;(Bε/2,S) .

Combining with (51) and (52) about ‖∇u‖C0 , we get

ε ‖w1‖Cα(Aε,S+) + ε1+α ‖u‖C0−(Aε,S)

+ ‖v‖C0−(Aε,S)

≥ CC (α) ε1+2α ‖u‖C1,α− (Bε/2,S)

.(53)

Covering Aε by radius-ε half balls Bε and radius-ε/2 full balls centeredon meridian Σ× ε/2 and then taking supremum on Aε, we derive

ε ‖w1‖Cα(Aε,S+) + ε1+α ‖u‖C0−(Aε,S)

+ ‖v‖C0−(Aε,S)

≥ CC (α) ε1+2α ‖u‖C1,α

− (Aε,S).(54)

Notice that here v is involved on the left side to control u, comparedto similar inequality (43) about v which needs no u. This is partlybecause in second order elliptic equations (39) of u and v, the inho-mogeneous term ∂x1w1 is “more discontinuous” than −∂−w1, for the


(weak) derivative is taken in x1 direction along which (the extended)w1 is discontinuous. Plugging these into (44) again, we obtain

C (α) ‖u‖C1,α− (Aε,S)

≤ ε−2α ‖w1‖Cα(Aε,S+) + ε−α ‖u‖C0−(Aε,S)

+ ε−(1+2α) ‖v‖C0−(Aε,S)

≤ ε−2α ‖w1‖Cα(Aε,S+) + C (p, λ) ‖w1‖C0(Aε,S)ε−(

3p+2α

)

.(55)

The C1,α estimates (55) and (45) also hold when Aε is equipped witha warped product metric gAε,h with h (z) ∈ C∞ (Σ) such that 1

K ≤h (z) ≤ K, up to the constant factor K > 0 on the right hand sides of(55) and (45). This is because they are derived by C0 estimates (44),(46), and local Schauder estimates (42), (53) for u, v on half balls Bε (p)and Bε/2 (p) for p ∈ ∂Aε = (0, z) , (ε, z) |z ∈ Σ, while when ε→ 0, the

function h (z) |Bε(p) converges to constant function h (p) uniformly in C0

norm in Bε (p), and correspondingly the second order equations of u,v converge to those with constant coefficients as above. This is similarto the “frozen coefficient method” in Schauder theory of second orderelliptic equations. Finally, we estimate the right inverse bound of D inthe Schauder setting. By Theorem 13, the operator D is surjective, sofor any W = (w1, w2) we can write it as the sum of (w1, 0) and (0, w2),whose preimages V = D−1 (w1, 0) and D−1 (0, w2) exist. Therefore wecan reduce general cases to cases (i) and (ii). Combining the C1,α es-timates (34) in case (i) and (55) and (45) for u and v in case (ii), wehave

C (α, p, λ) ‖V ‖C1,α− (Aε,S)

≤ ε−(

3p+2α

)

‖W‖Cα(Aε,S),

where C (α, p, λ) = C (α) (1 + C (p, λ))−1 is independent on ε. Hencethe result. q.e.d.

Remark 15. (About the asymmetry of case (i) and case (ii)) In case(i), since v|∂Aε = 0, after odd reflections of v and even reflections of uand w2, the V = (u, v) is still a C1,α function on the domain Ak(ε)ε

and w2 is of class Cα on Ak(ε)ε, so we can use the Schauder estimateof the domain Ak(ε)ε, which has uniform ellipticity; in case (ii), sincew1|∂Aε 6= 0 in general, after odd reflections of v and w1, w1 is no longercontinuous on Ak(ε)ε , not to mention in C1,α. So we have to directlywork on the thin domain Aε (which lacks uniform ellipticity as ε → 0)for the Schauder estimates, with ε-dependent coefficients. The steps

there are from Lp, C1− 3p , C0 to C1,α.

We remark that when λ∂+ > 0 but λ∂− = 0 (i.e. ker ∂− 6= 0), therestill exists a right inverse Qε : Cα (Aε,S) → C1,α

− (Aε,S) of D with the


operator norm bound ‖Qε‖ ≤ Cε−(

3p+2α

)

. To prove this, one needs toconsider the restriction

D : L1,2− (Aε,S) ∩ (kerD)⊥ → L2 (Aε,S)

to construct the right inverse Qε, where “⊥” is the L2 orthogonal com-plement, and replace the constants

λD− = inf06=V ∈L1,2

− (Aε,S)

‖DV ‖2L2(Aε,S)

‖V ‖2L2(Aε,S)

,

λ∂− = inf06=V ∈L1,2

− (Σ,S+)

‖∂−V ‖2L2(Σ,S+)

‖V ‖2L2(Σ,S+)

in Theorem 12 by the constants

λD− = inf06=V ∈L1,2

− (Aε,S)∩(kerD)⊥

‖DV ‖2L2(Aε,S)

‖V ‖2L2(Aε,S)

,

λ∂− = inf06=V ∈L1,2

− (Σ,S+)∩(ker ∂−)⊥

‖∂−V ‖2L2(Σ,S+)

‖V ‖2L2(Σ,S+)

respectively to get the L2 estimate of V ; and in the C0 estimate of u incase (ii) of the above theorem, use the fact that

u ∈ (kerD)⊥ ⇒ u ∈(ker ∂−

)⊥

by integrating in the x1 direction. The details are left to readers.From now on, we fix p > 3 sufficiently large and 0 < α < 1 sufficiently

small such that

0 <3

p+ 3α <

1

2.

This will be necessary for the implicit function theorem in the nextsection.

4. Proof of the main theorem

Let C0 ⊂M be a compact coassociative submanifold. Suppose thatn is a normal vector field on C0 such that its corresponding self-dualtwo form

η0 = ιnΩ ∈ ∧2+ (C0)

is harmonic with respect to the induced metric. So η0 is actually asymplectic form on the complement of the zero set Z (η0) of η0 in C0.Furthermore,

(56) Jn (u) :=n

|n| × u


defines an almost complex structure Jn on the C0\Z (η0) . Since de-formations of coassociative submanifolds are unobstructed, we may as-sume that there is a one parameter family of coassociative submanifoldsϕ : [0, ε] × C0 −→M such that

∂ϕ

∂t

∣∣∣∣t=0

= n ∈ Γ(C0, NC0/M

).

For ε small, let

C : = [0, ε] × C0, C := ϕ (C) , and Ct := ϕ (t ×C0) .

Then C is diffeomorphic to C, and ϕ (t, ·) is an embedding for ∀t ∈ [0, ε].We remind the readers about the typefaces of our notations: both C andC are 5-dimensional, while each Ct is 4-dimensional. Let

gC := dt2 ⊕ g|C0

be the product metric on C and expC be the exponential map associatedto the metric gC.

In the remaining part of this article, we assume that η0 is nowherevanishing on C0, which implies that (C0, η0) is a symplectic four man-ifold, and all coassociative submanifolds Ct’s are mutually disjoint. Weare going to establish a correspondence between the regular Jn-holomorphiccurves Σ in C0 and the instantons in M with coassociative boundaryconditions.

Given such a Σ ⊂ C0, we denote

Aε := [0, ε] ×Σ and A′ε := ϕ (Aε) .

Then A′ε is close to being associative in the sense that

∣∣τ |A′ε∣∣ ≤ Kε for

some constant K depending on the geometry of the family Ct andM for small ε. This is due to the smooth dependence of τ (ϕ (t, z)) on(t, z) ∈ [0, ε]× Σ, and the fact that

τ |TA′ε|ϕ(0,Σ)= τ |TA′ε|Σ = 0,

since Σ is a Jn-holomorphic curve, the tangent space TΣ is closed underJn = n

|n|×, and TA′ε|Σ = TΣ⊕spann.We want to perturb A′

ε to become an honest associative submanifoldin M . In order to apply the implicit function theorem to obtain thedesired perturbation for A

′ε, we need the estimates for the linearized

problem to behave well as ε approaches zero. Notice that for small ε,the induced metric on A

′ε is close to being a warped product metric

gAε,h := h (z) dx21 + gΣ

on Aε, where h (z) = |n (z)|2 is the length squared of the normal vectorn. Namely,

(1−Kε) gAε,h ≤ ϕ∗gM ≤ (1 +Kε) gAε,h


for some uniform constant K. This is because for any vectors X,Y ∈T(t,z)Aε0 and t ∈ [0, ε0], the function G : [0, ε0]× TAε0 × TAε0 → R,

G (t,X, Y ) := gM (dϕ (t, z)X, dϕ (t, z)Y ) ,

is smooth with respect to (t,X, Y ), bilinear in (X,Y ), and

G(0, TAε0 |0×Σ, TAε0 |0×Σ

)

= gM(dϕ|0×Σ

(TAε0 |0×Σ

), dϕ|0×Σ

(TAε0 |0×Σ

))

= gAε0 ,h|0×Σ,

where in the last identity we have used that dϕ|0×Σ = id : TΣ → TΣ

and dϕ|0×Σ : ∂∂x1

→ n.For the warped product metric on Aε and the corresponding operator

D, the estimates for its inverse have been established in Theorem 14 inthe previous section. The above discussion indicates that the lineariza-tion of the instanton equation on A′

ε may be compared with D. We willfirst show they agree on Σ in the next subsection. Then the comparisonon A′

ε is a small perturbation from their agreement on Σ.

4.1. Geometry of Σ ⊂ C ⊂ M . We study the geometry of the Jn-holomorphic curves Σ in a G2-manifold M . Let C = ∪0≤t≤εCt be thefamily of coassociative manifolds in the previous subsection and Σ ⊂C = C0. We remind the readers that the C is a 5-dimensional submani-fold so its normal bundle inM has real rank 2, while C is a 4-dimensionalsubmanifold. We recall the following results in Lemma 3.2 of [17].

Proposition 16. 1) NΣ/C and NC/M |Σ are complex line bundleswith the almost complex structure Jn = n

|n|× on fibers.

2) NC/M |Σ ≃ ∧0,1C

(NΣ/C

)as complex line bundles, where L is the

conjugate of a complex line bundle L, and ∧0,1C

(NΣ/C

)= NΣ/C⊗C

∧0,1C

(T ∗Σ).

Note that the notations ∂Y,X, νX , and µX in [17] correspond to ourΣ, C,NΣ/C , and NC/M |Σ, respectively.

In the following we will use the notation NC/M |Σ rather than NC/M |Σ,but we emphasize that the isomorphism NC/M |Σ ≃ ∧0,1

C

(NΣ/C

)is com-

plex conjugate linear and should be regarded as an isomorphism betweenreal vector bundles. Note that we do NOT complexify NΣ/C , since it isalready a complex line bundle, and more importantly, we want to useits complex structure Jn = n

|n|× to interplay with the G2 geometry.

In the following proposition, we will show that NΣ/C and NC/M |Σ areHermitian line bundles, andN := NΣ/C⊕NC/M |Σ → Σ is a Dirac bundle(in the sense of Definition 5.2 in [23]), i.e. N is a left Clifford module overΣ with respect to the G2 multiplication ×, together with a Riemannian


metric 〈, 〉 and connection ∇N on N satisfying: (a) at each p ∈ Σ, forany σ1, σ2 ∈ Np and any unit vector e ∈ TpΣ, 〈eσ1, eσ2〉 = 〈eσ1, eσ2〉;(b) for any section ϕ of TΣ and section σ of N , ∇N (ϕ× σ) =

(∇TΣϕ

)×

σ + ϕ×∇Nσ.

Proposition 17. Let ∇ be the Levi-Civita connection of (M,g). Forany subbundle L → Σ0 of TM |Σ0 , let the induced connection ∇L of Lbe

∇L := πL ∇,where πL is the orthogonal projection of TM |Σ to subbundle L accordingto the metric g. Then,

1) For L = NΣ/C , NC/M |Σ or TΣ, the induced connection ∇L is

Hermitian, namely ∇LJn = 0.2) Let N = NΣ/C ⊕ NC/M |Σ. Then with respect to the G2 multipli-

cation × as the Clifford multiplication and the induced connection∇N , N is a Dirac bundle.

Proof. For each p ∈ Σ0, the subspace TpΣ0⊕spann(p) ⊂ TpM is as-sociative since Σ0 is Jn-holomorphic. By the same Cayley-Dickson con-struction in Theorem 9, we may choose orthonormal frame Wαα=1,...7

in a neighborhood Bε (p) ⊂ Σ of p satisfying standard ·,× relation asthe basis of ImO, such that

TΣ0 = span W2,W3 , W1 :=W2 ×W3,

W4 ∈ NΣ 0/C0, Wi+4 :=Wi ×W4 for 1 ≤ i ≤ 3,

NΣ 0/C0= span W4,W5 , and NC/M |Σ0 = span W6,W7 .

For i = 2, 3, we have

∇LWiJn = ∇L

Wi (W1 × |L) = πL (∇WiW1)×|L = 〈W1,∇WiW1〉W1×|L = 0,

where the third identity is because for L = TΣ = spanW2,W3, L =NΣ0/C = spanW4,W5 or L = NC/M |Σ0 = spanW6,W7, under crossproduct ×, only the W1 component of ∇WiW1 can preserve L, i.e.

W1 × L ⊂ L, and (Wk × L) ∩ L = 0 for 2 ≤ k ≤ 7

from octonion multiplication relation, and the last identity is because〈W1,∇WiW1〉 = 1

2∇Wi 〈W1,W1〉 = 0. Thus ∇Lv J = 0 for any v ∈ TΣ0.

Using the standard G2 multiplication relation of the above “good” frameWαα=1,...7, it is easy to check that N := NΣ/C ⊕ NC/M |Σ → Σ is aClifford module. To show N is a Dirac bundle, we first note that forany unit vector e ∈ TpΣ, e× is an isometry on N by the property of ×;second, for any section ϕ of TΣ and section σ of NΣ/C ⊕NC/M |Σ, withrespect to the induced connection ∇N on N we have

∇N (ϕ× σ) = πN (∇ϕ× σ + ϕ×∇σ) =(∇TΣϕ

)× σ + ϕ×∇Nσ,


where in the second identity, the first term is because for distinct i, j ∈4, 5, 6, 7,Wi×Wj ∈spanW1,W2,W3, and the second term is becausespanW1,W2,W3 is closed under ×. q.e.d.

It is well known that on a compact Kahler manifold with a Hermitianline bundle L, one can define a Dirac operator on the Dolbeault complexΩ0,∗C

(L) (cf. Proposition 3.67 of [6] or Proposition 1.4.25 of [31]). Takingthe Kahler manifold to be Σ and the Hermitian line bundle to be NΣ/C ,

we have the Dolbeault Dirac operator on NΣ/C ⊕ ∧0,1C

(NΣ/C

). On the

other hand, by the above lemma, NΣ/C ⊕NC/M |Σ is a Dirac bundle andhas a canonically associated Dirac operator. The following propositioncompares the two Dirac operators.

Proposition 18. The Dolbeault Dirac operator on NΣ/C⊕∧0,1C

(NΣ/C

)

agrees with the Dirac operator on NΣ/C ⊕ NC/M |Σ, for which the Clif-ford multiplication is the G2 multiplication × and the connection is theinduced connection from M .

Proof. Given any p ∈ Σ, we may further assume the “good” frameWαα=1,2,...7 on Bε (p) in Proposition 17 satisfies

(57) ∇TΣ0Wi

Wj (p) = ∇NΣ0/C

WiWk (p) = ∇NC/M |Σ0

WiWk+2 (p) = 0

for 2 ≤ i, j ≤ 3 and 4 ≤ k ≤ 5, where the ∇TΣ0 ,∇NΣ0/C0 , and ∇NC/M |Σ0

are orthogonal projections of the Levi-Civita connection ∇ on M toTΣ0, NΣ 0/C0

, and NC/M |Σ0 , respectively (with respect to the metric g).

To achieve (57), we can first require ∇Σ0WiWj (p) = ∇NΣ0/C

WiW4 (p) = 0

for 2 ≤ i, j ≤ 3, and then useWi+4 :=Wi×W4 for 1 ≤ i ≤ 3 to show allother covariant derivatives in (57) vanish at p. Let Z = 1√

2(W2 − iW3) ∈

TΣ1,0C

, Z = 1√2(W2 + iW3) ∈ TΣ0,1

C, and Z

∗=

W ∗2 −iW ∗

3√2

∈ T ∗Σ0,1C

,

then⟨Z

∗, Z⟩

= 1. We recall that the real bundle isomorphism f :

∧0,1C

(NΣ/C

)≃ NC/M |Σ from [17] (up to factor − 1√

2) in local frame was

given by(58)

f : ∧0,1C

(NΣ/C

)=W4 ⊗C T

∗Σ0,1 −→ W4 × TΣ1,0 = NC/M |Σ,W4 ⊗C Z

∗ 7→ − 1√2W4 × Z,

which is complex conjugate linear and independent on the choice ofW4 ∈ NΣ/C . So we have the map

Φ := id⊕ f : NΣ/C ⊕ ∧0,1C

(NΣ/C

)→ NΣ/C ⊕NC/M |Σ


whose action on the basis of (rank two real vector bundles) NΣ/C and

∧0,1C

(NΣ/C

)is the following:

Φ : NΣ/Cid→ NΣ/C , W4,W5 → W4,W5 ,

Φ : ∧0,1C

(NΣ/C

) f→ NC/M |Σ,W4 ⊗ Z

∗,W5 ⊗ Z

∗→ W6,W7 .(59)

This is because under f ,

W4 ⊗(W ∗

2 − iW ∗3√

2

)→ − 1√

2

W4 ×W2 −W4 × iW3√2

= −−W6 − (W1 ×W4)×W3

2=W6,

W5 ⊗(W ∗

2 − iW ∗3√

2

)= JnW4 ⊗

(W ∗

2 − iW ∗3√

2

)

→ −Jn (W6) = −W1 ×W6 =W7,

where the second row is from the tensor property of f and the last rowis because f : ∧0,1

C

(NΣ/C

)→ NC/M |Σ is complex conjugate linear. So

for the sections

U = V 4W4 + V 5W5 + V 6W4 ⊗ Z∗+ V 7W5

⊗ Z∗ ∈ Γ

(NΣ/M ⊕ ∧0,1

C

(NΣ/C

)),

V = V 4W4 + V 5W5 + V 6W6 + V 7W7 ∈ Γ(NΣ/C ⊕NC/M |Σ

),

we haveΦ (U) = V.

We compute the Dolbeault Dirac operator using the above frame. TheClifford multiplication c on the Dolbeault complex NΣ/C ⊕∧0,1

C

(NΣ/C

)

at p is (see Section 1.4.3 in [31])

c

(W2 + iW3√

2

): V 4W4 + V 5W5

√2e(Z

∗)→√2(V 4W4 ⊗ Z

∗+ V 5W5 ⊗ Z

∗),

c

(W2 − iW3√

2

): V 6W4 ⊗ Z

∗+ V 7W5 ⊗ Z

∗ −√2i(Z)→

√2(−V 6W4 − V 7W5

),

where e(Z

∗)is the wedge by Z

∗and i

(Z)is the contraction by Z.

Using (57) at p and complex linearity, we have

∇(

W2+iW3√2

)

(V 4W4 + V 5W5

)(p)

=1√2

(V 42 W4 + V 4

3 JnW4 + V 52 W5 + V 5

3 JnW5

)

=1√2

((V 42 − V 5

3

)W4 +

(V 52 + V 4

3

)W5

)


and

c

(W2 + iW3√

2

) ∇(

W2+iW3√2

)

(V 4W4 + V 5W5

)(p)

=(V 42 − V 5

3

)W4 ⊗ Z

∗+(V 52 + V 4

3

)W5 ⊗ Z

∗.

Similarly, we have

∇(

W2−iW3√2

)

(V 6W4 ⊗ Z

∗+ V 7W5 ⊗ Z

∗)(p)

=1√2

[(V 62 + V 7

3

)W4 ⊗ Z

∗+(−V 6

3 + V 72

)W5 ⊗ Z

∗]

and

c

(W2 − iW3√

2

) ∇(

W2−iW3√2

)

(V 6W4 ⊗ Z

∗+ V 7W5 ⊗ Z

∗)(p)

= −(V 62 + V 7

3

)W4 −

(−V 6

3 + V 72

)W5.

The Dolbeault Dirac operator ∂ on NΣ/M ⊕ ∧0,1C

(NΣ/M

)is defined by

∂ = c

(W2 + iW3√

2

) ∇(

W2+iW3√2

) + c

(W2 − iW3√

2

) ∇(

W2−iW3√2

),

so at p we have

∂U (p) = −(V 62 + V 7

3

)W4 +

(V 63 − V 7

2

)W5

+(V 42 − V 5

3

)W4 ⊗ Z∗ +

(V 52 + V 4

3

)W5 ⊗ Z∗,

and

Φ(∂U (p)

)= −

(V 62 + V 7

3

)W4 +

(V 63 − V 7

2

)W5

+(V 42 − V 5

3

)W6 +

(V 52 + V 4

3

)W7.

On the other hand, the twisted Dirac operator on N := NΣ/C ⊕NC/M |Σis

(60) D :=W2 ×∇NW2

+W3 ×∇NW3.

Using (57) at p, we have

DV (p) =(W2 ×∇N

W2+W3 ×∇N

W3

) (V 4W4 + V 5W5 + V 6W6 + V 7W7

)

=(V 42 W2 ×W4 + V 4

3 W3 ×W4

)+(V 52 W2 ×W5 + V 5

3 W3 ×W5

)

+(V 62 W2 ×W6 + V 6

3 W3 ×W6

)+(V 72 W2 ×W7 + V 7

3 W3 ×W7

)

= −(V 62 + V 7

3

)W4 +

(V 63 − V 7

2

)W5

+(V 42 − V 5

3

)W6 +

(V 43 + V 5

2

)W7.(61)

Therefore

Φ(∂U (p)

)= DV (p) .


Note that Dirac operators are independent on the choice of orthonormalbasis at p, and our p ∈ Σ is arbitrary, so Φ

(∂U)= DV on Σ. The proof

of the proposition is completed. q.e.d.

We now make the connection to Subsection 3.1. Taking L = NΣ/C in

that subsection, then, S+ = NΣ/C and S− = ∧0,1

C

(NΣ/C

). Our assump-

tion that Σ is regular implies dimH0,1

∂

(NΣ/C

)= 0, for H0,1

∂

(NΣ/C

)

corresponds to the cokernel of ∂. By the Dolbeault isomorphism, wehave

dimH1(Σ, NΣ/C

)= dimH0,1

∂

(NΣ/C

)= 0.

Since the dimension for the Seiberg-Witten moduli is 0, by the equiva-lence to Gromov-Witten moduli we have

dimH0(Σ, NΣ/C

)= dimH1

(Σ, NΣ/C

)= 0,

for we only count Jn-holomorphic curves Σ of index 0. Hence for

∂ : Ω0(NΣ/C

)→ Ω0,1

(NΣ/C

)

∂∗: Ω0,1

(NΣ/C

)→ Ω0

(NΣ/C

)

in Subsection 3.1, ker ∂ and ker ∂∗are trivial. Moreover, by Theorem

13, the linear operator

D : L1,2− (Aε,S) → L2 (Aε,S)

is one-to-one and onto.

4.2. Linearization of instanton equation and comparison withthe operator D. To prove the main theorem (Theorem 27), we willconstruct a map

Fε : Cm,α−

(NA′

ε/M

)→ Cm−1,α

(NA′

ε/M

)

such that the solution to the equation Fε (V ) = 0 will give rise to anassociative submanifold (instanton) with boundary lying on C0 ∪ Cε.The spaces Cm,α

(NA′

ε/M

)and Cm,α

−(NA′

ε/M

)are defined by

Cm(NA′

ε/M

):=

V ∈ Γ

(NA′

ε/M

)∣∣ ‖V ‖Cm

(

NA′ε/M

) < +∞,

Cm−(NA′

ε/M

):=

V ∈ Γ

(NA′

ε/M

)∣∣∣∣∣

‖V ‖Cm

(

NA′ε/M

) < +∞, and

V |ϕ(0×Σ) ⊂ TC0, V |ϕ(ε×Σ) ⊂ TCε.

.

We construct a three dimensional submanifold A′ε = ϕ (Aε) ⊂ M by

flowing Σ along with Ct. For the spinor bundle S →Aε, we will constructan exponential-like map exp : S →M , with the following properties ofthe differential dexp|Aε on 0 × Σ:

1) on fiber directions of S, dexp|0×Σ = (id, f) : NΣ/C⊕∧0,1C

(NΣ/C

)→

NΣ/C ⊕NC/M |Σ, where f : ∧0,1C

(NΣ/C

)→ NC/M |Σ is the real vec-

tor bundle isomorphism in (58);


2) on base directions of S, dexp|0×Σ = id : TΣ → TΣ and dexp|0×Σ :∂

∂x1→ n (z) ;

3) on the boundary 0, ε × Σ of Aε, exp|0×Σ (S+ ⊕ 0) ⊂ C0, and

exp|ε×Σ (S+ ⊕ 0) ⊂ Cε.

The construction of the map exp is somewhat technical. To keep themain flow of our paper, we postpone it to the appendix.

To make the linear theory developed in the previous section applica-ble, we compare the linearization DFε (0) with the operator D in Section3.1 by the following diagram to get the ε-dependent bound of its rightinverse:

Cm,α− (Aε,S)

D−→ Cm−1,α− (Aε,S)

dexp ↓ ↓ dexpCm,α−

(NA′ε/M

) DFε(0)−→ Cm−1,α(NA′ε/M

)

Now we define the nonlinear map Fε with the important property thatelements in F−1

ε (0) with small norm correspond to associative subman-ifolds inM near A′

ε for small ε. Given any C0∪Cε, we modify the metricg near C0 and Cε to make them totally geodesic. We denote this newsmooth metric by gε, and we make gε C

1-continuously depend on ε inour construction. Let expgε be the exponential map of gε. Then expgε

has the following properties:

1) For sections V of NA′ε/M with C0 norm smaller than a fixed con-stant δ0 (depending on the uniform injectivity radius of the familyof metrics gε0≤ε≤ε0

),

expgε V : A′ε →M

is a smooth embedding.2) For V ∈ Cm,α

−(NA′

ε/M

), let

(62) Aε (V ) := (expgε V )(A′ε

).

Then Aε (V ) is a submanifold ofM nearby A′ε satisfying the bound-ary condition

(63) ∂Aε (V ) ⊂ C0 ∪ Cε.

3) For small t ≥ 0, the family of embeddings expgε (tV ) : A′ε → Msatisfies

(64)d

dt

∣∣∣∣t=0

expgε (tV ) = V.

Next we define Fε : Cm,α−

(A′ε, NA′ε/M

)→ Cm−1,α

(A′ε, NA′ε/M

),

(65) Fε (V ) = ∗A′ε ⊥A′ε (TV (expgε V )∗ τ) ,

where

1) (expgε V )∗ pulls back the differential form part of τ,


2) TV : Texpgεp (tV )M → TpM pulls back the vector part of τ by the

parallel transport with respect to g along the path expgεp (tV ) ,3) ⊥A′ε : TM |A′ε → NA′ε/M is the orthogonal projection with respect

to g,4) ∗A′ε : Ω3 (A′ε) → Ω0 (A′ε) is the quotient by the volume form dvolA′ε

induced from g.

We stress that gε is only used to construct a map expgε : NA′ε/M →Msatisfying the coassociative boundary condition (63) and derivative con-dition (64). For our covariant derivatives, parallel transport, orthogonalprojection, and volume form, we still use the original metric g.

To better understand Fε, we let

Pε := ∗A′ε ⊥A′ε : Γ(TM |A′ε

)⊗ Ω3

(A′ε

)→ Γ

(NA′ε/M

),

and

(66) F gε (V ) := TV (expgε V )∗ τ.

Then

Fε (V ) = Pε F gε (V ) .

At any p ∈ A′ε, both ⊥A′ε and ∗A′ε are linear operators between finite

dimensional spaces, so ‖Pε‖ ≤ C, where the constant C only dependson ϕ and is uniform for all 0 < ε ≤ ε0. We notice that Pε does notinvolve V , so the essential part of Fε (V ) is F gε (V ).

By Proposition 8, if A′ε is sufficiently close to being associative, thenthere exists δ > 0 such that for ‖V ‖

C1,α−

(

A′ε,NA′

ε/M

) < δ,

Fε (V ) = 0 ⇔ F gε (V ) = 0 ⇔ Aε (V ) associative.

We also note that ∂Aε (V ) ⊂ C0 ∪Cε for V ∈ C1,α−(A′

ε, NA′ε/M

).

Before we compute F gε′ (0)V , we observe the following useful fact.The original exponential map expg : NA′ε/M → M does not satisfy thecoassociative boundary condition (63), but satisfies the same derivativecondition (64):

d

dt

∣∣∣∣t=0

expg (tV ) = V =d

dt

∣∣∣∣t=0

expgε (tV ) .

For smooth maps f : A′ε →M and a smooth form τ , the nonlinear map

Γ : f → f∗τ is differentiable with respect to f . Therefore

(67)d

dt

∣∣∣∣t=0

(expg (tV ))∗ τ = Γ′ (0)V =d

dt

∣∣∣∣t=0

(expgε (tV ))∗ τ.

Recall that the F (V ) (6) defined in our proof of McLean’s theorem is

F (V ) := TV (expg V )∗ τ,


where the parallel transport TV is with respect to g along the geodesicexpg (tV ). F gε (V ) and F (V ) are different nonlinear maps, but (67) saysthat

(68) F gε′ (0)V = F ′ (0)V .

We give an alternative proof of (68) in the following lemma. The proofgives more information of F gε′ (0)V when there is a good frame field forthe vector-valued form τ .

Lemma 19. For any V ∈ C1,α−(NA′

ε/M

), F gε′ (0)V = F ′ (0)V .

Proof. For any p ∈ A′ε, we choose a frame field Wa1≤α≤7 in its

neighborhood B ⊂ A′ε, and then extend V and Wa1≤α≤7 to M by

the parallel transport with respect to g along the curve expgεp (tV ). Wewrite τ = ωα ⊗Wα in the neighborhood of p in M , following Einstein’ssummation convention. Similar to (12), we compute

F gε′ (0)V = d (iV ωα)⊗Wα + iV dω

α ⊗Wα + ωα ⊗∇VWα,

where the covariant derivative ∇ is with respect to g, and we have usedthat d

dt |t=0 expgε (tV ) = V . By the parallel property of τ and Wα with

respect to g, we have

iV dωα ⊗Wα = 0 = ωα ⊗∇VWα

(also see Remark 10, item 3). The first term d (iV ωα) |A′

εonly depends

on the restriction of V and ωα on A′ε (Remark 10, item 1). Therefore,

F gε′ (0)V = d (iV ωα)⊗Wα|A′

ε= F ′ (0)V.

q.e.d.

Remark 20. To apply the implicit function theorem to Fε (V ), inthe remaining part of our paper we will only need the estimate of F ′

ε (0),and the quadratic estimate (80) of F ′

ε (V ). Because of the above lemma,to compute F ′

ε (0) = Pε F gε′ (0), we can replace the metric gε by g in(66). This will simplify the exposition in many places. For our quadraticestimate (80), the proof uses no feature of the G2 metric g and is validfor any Riemannian metrics, including gε. The constant C in (80) isuniform for gε0≤ε≤ε0

, since this is a compact family of metrics C1-continuously depending on ε. So although we used gε in the definitionof Fε (V ) (65), from now on we will pretend gε is g in Fε (V ), and wewill simply write expg as exp.

Proposition 21. For any section V1 of S and section V2 := dexp ·V1of NA′ε/M , we have∥∥F ′ (0)V2 − (dexp DV1)⊗ dvolA′

ε

∥∥Cα

(

A′ε,NA′

ε/M

) ≤ Cε1−α ‖V1‖C1,α(Aε,S),


and∥∥F ′

ε (0)V2 − (dexp) DV1∥∥Cα

(

A′ε,NA′

ε/M

) ≤ Cε1−α ‖V1‖C1,α(Aε,S),

where D is the operator on S in the linear model, and the constant C isuniform for all ε.

Proof. For each p = ϕ (0, z) in Σ0 := ϕ (0 × Σ) ⊂ A′ε, we choose

“good” frame Wαα=1,2,...7 on Bε (z) ⊂ Σ as before. Then we extend

the frame to Uε (p) := ϕ ([0, ε] ×Bε (z)) such that

Wα (ϕ (t, z)) = Tγ ·Wα (ϕ (0, z)) for α = 1, 2, . . . 7,

where Tγ is the parallel transport along the path γ := ϕ ([0, t] × z)by the Levi-Civita connection on M . We further extend Wαα=1,2,...7

to a tubular neighborhood of Uε (p) ⊂ M by parallel transport in fiberdirections of NA′

ε/M. Both F ′ (0)V2 and DV1 are globally defined on A′

ε

and Aε respectively. We are going to compare them in Uε (p) using the“good” frame. We write τ = ωα ⊗ Wα in the neighborhood of p. Bythe parallel property of Wαα=1,2,...7, the 3-forms ωα are similar to the

standard ones in ImO, in the sense that dxi is replaced by (Wi)∗ for

i = 1, 2, . . . 7. We still have the formula (12) at q:

F ′ (0)V2 (q) = d (iV2ωα)⊗Wα (q)

+ iV2dωα ⊗Wα (q) + ωα ⊗∇V2Wα (q) .(69)

The first term d (iV2ωα) ⊗ Wα (q) is the principal symbol part of the

differential operator F ′ (0). We claim that

(70) d (iV2ωα)⊗Wα = (dexp · DV1)⊗ dvolA′

ε+ E (q)V1

where E (q) is a smooth tensor on Uε (p) with ‖E‖C1(Uε(p))≤ C3ε. To

see this, we first compute d (iV2ωα)⊗Wα. For section V2 of NA′

ε/M, we

writeV2 (q) = Σ7

α=4φα (q)Wα (q) .

Similar to our proof of McLean’s theorem, we have

(71) d (iV2ωα)⊗Wα (q) |TqA′

ε= DV2 (q) dvolA′

ε+ E1 (q)V2

where

DV2 (q) = −(φ51 + φ62 + φ73

)W4 +

(φ41 + φ63 − φ72

)W5

+(φ42 − φ53 + φ71

)W6 +

(φ43 + φ52 − φ61

)W7,(72)

andφki (q) := dφk (q) (Wi) .

The reason is the following: From our construction of Wα, ∇WiWj (p) =0 for 1 ≤ i, j ≤ 7, and ωα are similar to the standard ones in ImO. Thuswhen q = p,

d (iV2ωα)⊗Wα (p) |TpA′

ε= DV2 (p) dvolA′

ε.


If q is ε-close to p, then in (71) the error term E1 (q) is of order ε inC1 norm, because ∇WiWj (q) = o (ε) in C1 and spanWα (q)α=1,2,3 has

ε-order deviation from TqA′ε in C1. We want to show

(73) DV2 (q) = dexp [(h−1/2 (z) e1 ·

∂

∂x1+ ∂

)V1

]+ E2 (q)V2,

where the error term E2 (q) is of order ε in C1 norm, and the ∂ is

the Dolbeault Dirac operator on NΣ/C ⊕ ∧0,1C

(NΣ/C

). Similar to our

argument for E1 (q), it is enough to show

DV2 (p) = dexp|0×Σ [(h−1/2 (z) e1 ·

∂

∂x1+ ∂

)V1 (0, z)

].

We observe that

DV2 (p) = DV2 (p)− φ51W4 + φ41W5 + φ71W6 − φ61W7

from (72) and (61), whereD is the twisted Dirac operator (60) ofNΣ/C⊕NC/M |Σ over Σ in the previous subsection. By Proposition 18, we have

dexp|0×Σ · ∂V1 = Φ(∂V1

)= DV2.

So it is enough to prove, at p = ϕ (0, z), that(74)

dexp[h−1/2 (z) e1 ·

∂

∂x1V1 (0, z)

]=(−φ51W4 + φ41W5 + φ71W6 − φ61W7

)(p) .

From

V2 (q) = Σ7α=4φ

α (q)Wα (q) = Σ7α=4φ

α (q) · TγWα (ϕ (0, z))

we get

V1 (t, z) = Σ7α=4φ

α (ϕ (t, z)) · (dexp)−1 TγWα (ϕ (0, z))

= Σ7α=4φ

α (ϕ (t, z)) · Φ−1Wα (ϕ (0, z)) +E3 (q)V2,

whereΦ−1Wα (ϕ (0, z))

7α=4

is regarded as a frame of S →Aε that is

invariant along the t direction, and E3 (q) is of order ε in C1 norm. The

second identity of V1 (t, z) is because the maps dexp and Tγ , at t = 0, aremaps Φ and the identity map respectively on spanWα4≤α≤7, and for

0 ≤ t ≤ ε we have the error term E3 (q) of desired order by smoothnessof ϕ, φα, and Wα. Let

ψα (x1, z) := φα (ϕ (x1, z)) , and ψα1 (x1, z) =

∂

∂x1ψα (x1, z) .

We have

e1 ·∂

∂x1

[Σ7α=4ψ

α (x1, z) · Φ−1Wα (ϕ (0, z))]

= Φ−1[W1 × Σ7

α=4ψα1 (x1, z) ·Wα (ϕ (0, z))

]

= Φ−1[−ψ5

1W4 + ψ41W5 + ψ7

1W6 − ψ61W7

].(75)


By the chain rule, for 4 ≤ α ≤ 7,

(76) ψα1 (0, z) = ∇W1φ

α (ϕ (0, z)) · h (z)12 = h (z)

12 φα1 (ϕ (0, z))

where the factor h (z)12 is from

⟨W1,

ddtϕ (t, z) |t=0

⟩= h (z)

12 . Comparing

(74) and (75), and noticing (76), we have

dexp|0×Σ DV1 = DV2.Therefore, for q that is ε-close to p, claim (70) is proved. So we get

∥∥d (iV2ωα)⊗Wα − (dexp DV1)⊗ dvolA′

ε

∥∥Cα(Uε(p))

= ‖E (q)V2‖Cα(Uε(p))

≤ C1ε1−α ‖V2‖C1,α(Uε(p))

,(77)

where the constant C1 is uniform for all ε. The remaining term

BV2 := iV2dωα ⊗Wα + ωα ⊗∇V2Wα

is a 0-th order linear operator on V2, where B = B (q) is a smooth tensoron A′

ε. Since ∇τ = 0, the same as in our proof in Theorem 9, we have

iW dωα (p) = ∇WWα (p) = 0 for all W ∈ NA′

ε/M(p) ,

so we have B (p) |NA′ε/M

(p) = 0. Since dist(p, q) ≤ C0ε for some uniform

constant C0, and NA′ε/M

is a smooth fiber bundle over A′ε, we conclude

that the operator norm∥∥∥B (q) |NA′

ε/M(q)

∥∥∥ ≤ C2ε

for some uniform constant C2. From this it is easy to prove

(78) ‖BV2‖Cα(Uε(p))≤ C3ε

1−α ‖V2‖C1,α(

A′ε,NA′

ε/M

) .

The constant C3 is uniform for all ε, by the compactness of Σ and finitecovering of A′

ε by Uε (p). Putting (78) and (77) in (69), we have∥∥F ′ (0)V2 − (dexp · DV1)⊗ dvolA′

ε

∥∥Cα(Uε(p))

≤ C4ε1−α ‖V2‖C1,α

(

A′ε,NA′

ε/M

)

≤ C5ε1−α ‖V1‖C1,α(Aε,S)

where the constants C4 and C5 are uniform for all ε. Using finitely manyUε (p) covering A

′ε and then taking supremum, we have

(79)∥∥F ′ (0)V2 − (dexp · DV1)⊗ dvolA′ε

∥∥Cα(A′

ε)≤ Cε1−α ‖V1‖C1,α(Aε,S)

.

Applying Pε on the left-hand side of the above inequality, and noticingthat (dexp) |0×Σ DV1 = DV2 is a section of NA′

ε/M, we have

Pε ((dexp) DV1 (t, z)⊗ dvolA′

ε

)= (dexp) DV1 (t, z) + E3 (q)V1,


where E3 (q) is of order ε in C1 norm, because at q = ϕ (t, z) the vec-tor (dexp) DV1 (t, z) may have a component of order ε in C1 normorthogonal to NA′

ε/M(q). So we get

∥∥F ′ε (0)V2 − (dexp) DV1

∥∥Cα

(

A′ε,NA′

ε/M

)

≤∥∥Pε ·

(F ′ (0)V2 − (dexp) DV1 ⊗ dvolA′

ε

)∥∥Cα

(

A′ε,NA′

ε/M

)

+ ‖E3 (q)V1‖Cα(A′ε)

≤ Cε1−α ‖V1‖C1,α(Aε,S)

where C is a uniform constant independent on ε. q.e.d.

Proposition 22. There exists a right inverse Qεtrue

of F ′ε (0), such

that∥∥∥Qε

true∥∥∥ ≤ Cε

−(

3p+2α

)

, where the constant C is uniform for all

0 < ε ≤ ε0.

Proof. From Theorem 14, for operator D on spinor bundle S over Aε,

there is a right inverse Qε of D such that ‖Qε‖ ≤ Cε−(

3p+2α

)

. Let

D = dexp D (dexp)−1 ,

Qε = dexp Qε (dexp)−1 ,

then∥∥∥Qε

∥∥∥ ≤ ‖dexp‖ · Cε−(

3p+2α

)

·∥∥∥(dexp)−1

∥∥∥ ≤ Cε−(

3p+2α

)

,

and

F ′ε (0) Qε − id =

(F ′ε (0)− D + D

)Qε − id

=(F ′ε (0)− D

)Qε + DQε − id

=(F ′ε (0)− D

)Qε,

where the last identity is because DQε = dexp DQε (dexp)−1 = id.

From the previous proposition,∥∥∥(F ′ε (0)− D

)∥∥∥ ≤ Cε1−α. Therefore

∥∥∥F ′ε (0) Qε − id

∥∥∥ ≤∥∥∥(F ′ε (0)− D

)∥∥∥∥∥∥Qε

∥∥∥

≤ Cε1−α · Cε−(

3p+2α

)

<1

2

when ε is sufficiently small, by our assumption that 1−(3p + 3α

)> 0.

So Qε is an approximate right inverse of F ′ε (0). The true right inverse


Qtrue of F ′ε (0) is Q

true = Qε

(F ′ε (0) Qε

)−1and

∥∥∥Qtrue∥∥∥ ≤

∥∥∥Qε

∥∥∥∥∥∥∥(F ′ε (0) Qε

)−1∥∥∥∥ ≤ Cε

−(

3p+2α

)

· 2 ≤ Cε−(

3p+2α

)

.

q.e.d.

4.3. Quadratic estimates.

Proposition 23. There exists δ0 > 0 such that for all sections V0, Vof NA′

ε/Mthat ‖V0‖C1,α

(

A′ε,NA′

ε/M

) < δ0, we have

∥∥F ′ε (V0)V − F ′

ε (0)V∥∥Cα

(

A′ε,NA′

ε/M

)

≤ C ‖V0‖C1,α(

A′ε,NA′

ε/M

) ‖V ‖C1,α

(

A′ε,NA′

ε/M

)

,(80)

where the constant C is independent on ε.

Proof. Because Fε (V ) = Pε F (V ) and Pε is a bounded linear oper-ator independent on V , it is enough to prove the quadratic estimate ofF (V ), namely

∥∥F ′ (V0)V − F ′ (0)V∥∥Cα

(

A′ε,NA′

ε/M

)

≤ C ‖V0‖C1,α(

A′ε,NA′

ε/M

) ‖V ‖C1,α

(

A′ε,NA′

ε/M

)

.

For any p = ϕ (t, z) on A′ε , we choose a normal frame field Wαα=1,2,...,7

in its neighborhood Bδ0 (p) where τ = ωα⊗Wa, with δ0 = the injectivityradius of M . For the point q := expp V0 and if |V0| < δ0, then we canassume this neighborhood covers q. Let the submanifold

Aε (V ) =: (expV )(A′

ε

),

which is similar to (62), except that the section V here is in NA′ε/M

instead of S, and the exp : NA′ε/M

→ M is the exponential map on M .For the family of embeddings of submanifolds

ψt := exp (V0 + tV ) : Aε → Aε (V0 + tV ) ⊂M,

we have

ψ0 (Aε) = Aε (V0) ,

d

dt

∣∣∣∣t=0

ψt (q) = (d expp V0)V (p) := V1 (q)


for any p ∈ A′ε = Aε (0). Arguing as in Theorem 9, from ∇τ = 0 we

have ∇Wα (q) = dωα (q) = 0. We have

F ′ (V0)V (p)

=d

dt

∣∣∣∣t=0

F (V0 + tV ) (p)

=d

dt

∣∣∣∣t=0

[exp (V0 + tV )∗ ωα (p)⊗ TV0+tVWα

(expp (V0 + tV )

)]

=d

dt

∣∣∣∣t=0

(exp (V0 + tV )∗ ωα)⊗Wα (p)

+ (expV0)∗ ωα (p)⊗ d

dt

∣∣∣∣t=0

TV0+tVWα

(expp (V0 + tV )

),(81)

where in the last equality we have used TV0+tVWα

(expp (V0 + tV )

)=

Wα (p) by the parallel property of Wα. For the first term of (81), by theproperty of the exponential map, namely d expq (0) = idTqM , we have

expp (V0 + tV ) = expq (tV1 + t |V | β (V0,V, t)) expp V0

where V1 =(d expA′

εV0

)V is a vector field on Aε (V0), β (V0,V, t) and

β′ (V0,V, t) are uniformly bounded, and limt→0 β (V0,V, t) = 0. Therefore

d

dt

∣∣∣∣t=0

exp (V0 + tV )∗ ωα (p)

=d

dt

∣∣∣∣t=0

(expA′

εV0

)∗expAε(V0) (tV1 + t |V | β (V0,V, t))∗ ωα

=(expp V0

)∗LV1ω

α

=(expp V0

)∗(d (iV1ω

α) + iV1dωα) ,

where in the third equality we have used that limt→0 β (V0,V, t) = 0. Forthe second term of (81), by the Gauss-Bonnet formula, which comparesthe parallel transports along different paths by curvature integration,we have

TV0+tVWα

(expp (V0 + tV )

)

= TV0 TtV1+t|V |β(V0,V,t)Wα

(expq (tV1 + t |V |β (V0,V, t))

)

+

∫

∆(V0,tV )R (σ) dσWα (p) ,

where R (σ) is the Ricci curvature tensor, ∆ (V0,tV ) is the 2-dimensionalgeodesic triangle with the vertices p, expp V0, and expp (V0 + tV ), and σ


is the area element. Therefore

d

dt

∣∣∣∣t=0

TV0+tVWα

(expp (V0 + tV )

)

= TV0 d

dt

∣∣∣∣t=0

TtV1Wα

(expq tV1

)+

d

dt

∣∣∣∣t=0

∫

∆(V0,tV )R (σ) dσWα (p) ,

where in the first term we have dropped the higher order termt |V | β (V0,V, t) since it is irrelevant for the derivative at 0. If we denote

B (V0, V ) :=d

dt

∣∣∣∣t=0

∫

∆(V0,tV )R (σ) dσ,

by the area formula of ∆ (V0,tV ) it can be shown that

|B (V0, V )Wα (p)| ≤ C5 |V0| |V | |Wα| (p) ,

where C5 is a constant only depending on (M,g). Plugging these into(81), we have

F ′ (V0) |Aε(0)V (p)

= (expV0)∗ LV1ω

α ⊗Wα (p) + (expV0)∗ ωα ⊗ TV0∇V1Wα (q)

+B (V0, V )Wα (p)

= ((expV0)∗ ⊗ TV0) [LV1ω

α ⊗Wα (q) + ωα (q)⊗∇V1Wα (q)]

+B (V0, V )Wα (p)

= ((expV0)∗ ⊗ TV0) [(d (iV1ω

α) + iV1dωα)⊗Wα (q)

+ωα (q)⊗∇V1Wα (q)]

+B (V0, V )Wα (p) .(82)

Here we have used that TV0Wα (q) = Wα (p). Equation (82) can berewritten as

F ′ (V0) |Aε(0)V (p) = ((expV0)

∗ ⊗ TV0) [F ′ (0) |Aε(V0)V1 (q)

]

+B (V0, V )Wα (p) ,

which means the derivative F ′ (V0) on Aε (0) can be expressed by thederivative F ′ (0) on Aε (V0) via the transform (expV0)

∗⊗TV0 , up to thecurvature term B (V0, V )Wα (p). If V0 = 0, then q = p and V1 = V ,together with

iV dωα (p) = ∇VWα (p) = B (0, V ) = 0,

the formula (82) is simplified as

F ′ (0) |A′εV (p) = d (iV ω

α)⊗Wα (p) .


Therefore

F ′(V0)V (p)− F ′ (0)V (p)

= [(expV0)∗ d(iV1ω

α)− d(iV ωα)] |Aε(0)

⊗Wα(p)+

(83)

((expV0)∗ ⊗ TV0) [iV1dω

α ⊗Wα + ωα ⊗∇V1Wα] (p) +B(V0, V )Wα(p).(84)

For the second line (84), it is a 0-th order linear differential operator onV ∈ C1,α

(Γ(NA′

ε/M

)), where

V1 = (d exp V0) |Aε(0)V = E (V0)V.

We notice that for the linear operator

H : C1,α(Γ(NA′

ε/M

))→ Cα

(Γ(Hom

(TM |A′

ε, TM |Aε(V0)

)))

that sends

V0 → ((expV0)∗ ⊗ TV0)

[iE(V0)(·)dω

α ⊗Wα + ωα ⊗∇E(V0)(·)Wα

]

+B (V0, ·)Wα,

H is a bounded operator since the terms expV0, (exp V0)∗, TV0 , E(V0),

andB(V0, ·) as elements in Cα are differentiable for V0 ∈ C1,α(Γ(NA′ε/M

))with bounded Frechet derivatives. The bound of the derivatives is uni-form on ε, so ‖H‖ is uniformly bounded. Since H(V0) = 0 when V0 = 0,we have

‖H (V0,V )‖Cα ≤ C8 ‖V0‖C1,α(A′ε)‖V ‖C1,α(A′

ε).

For the first line (83), [(expV0)∗ d (iV1ω

α)− d (iV ωα)] is a first order

linear differential operator on V(V1 = (d exp V0) |Aε(0)V

), and only in-

volves the covariant derivatives of V . In the next lemma we will show

‖(expV0)∗ d (iV1ωα)− d (iV ω

α)‖Cα(A′ε)≤ C7 ‖V0‖C1,α(A′

ε)‖V ‖C1,α(A′

ε).

Combining the two lines we get∥∥F ′ (V0)V − F ′ (0)V

∥∥Cα

(

A′ε,NA′

ε/M

)

≤ C ‖V0‖C1,α(

A′ε,NA′

ε/M

) ‖V ‖C1,α

(

A′ε,NA′

ε/M

)

.

q.e.d.

Lemma 24. For any V0, V ∈ Γ(NA′

ε/M

)with ‖V0‖C1,α(A′

ε)≤ δ0, we

have


α)‖Cα(A′ε)≤ C7 ‖V0‖C1,α(A′

ε)‖V ‖C1,α(A′

ε),

where the constants δ0 and C7 are independent on ε.


Proof. Consider the 3 linear maps from TpM to TqM , where q =expp V0 :

PalV0 : TpM → TqM, parallel transport along the geodesicexpp tV0 for 0 ≤ t ≤ 1,

d (expV0) : TpM → TqM, tangent map of the diffeomorphismexpV0 : A

′ε → Aε (V0) ,

d(expp

)(V0) : TV0 (TpM) → TqM, tangent map of the exponential map

expp : TpM →M,

where in the last one we identify TV0 (TpM) ≃ TpM . Note that fordiffeomorphism d (expV0) we need a vector field V0 on A′

ε, but ford(expp

)(V0) we only need V0 ∈ TpM , so they are essentially differ-

ent maps. But the 3 maps are very close to each other when V0 is small.More precisely, for any V0, V ∈ Γ

(NA′

ε/M

)with ‖V0‖C1,α(A′

ε)≤ δ0, we

have comparison of the maps as in the following:

d (expV0)V = PalV0V + h1 (V0)V,

d(expp

)(V0)V = d (expV0)V + h2 (V0)V,

where for each i = 1, 2, the error term

hi (·) : C1,α(Γ(NA′

ε/M

))→ Cα

(Γ(Hom

(TM |A′

ε, TM |Aε(V0)

)))

is differentiable with respect to the variable V0, and

hi (V0) = 0 for V0 = 0.

This is because for any fixed p, PalV0(p) and d(expp

)(V0) smoothly de-

pend on V0, and for the compact family p ∈ A′ε, PalV0 and d

(expp

)(V0)

inherit the smooth dependence on V0 ∈ C1,α(Γ(NA′

ε/M

)). For d (expV0),

since

d (expV0) (x) = F1 (x, V0) dx+ F2 (x, V0) dV0 (x)

where F1 (x, y) = ∂∂x expx y and F2 (x, y) = ∂

∂y expx y are bounded on

Aε, we see d (expV0) ∈ Cα smoothly depends on V0 ∈ C1,α(Γ(NA′

ε/M

))

with bounded Frechet derivative. This especially implies that

(85) ‖hi (V0)V ‖Ca ≤ C8 ‖V0‖C1,α(A′ε)‖V ‖C1,α(A′

ε)

for i = 1, 2. We have

V1 = d (expV0)V + h2 (V0)V,

d (expV0)Wα (p) =Wa (q) + h1 (V0)Wα (p) ,

(expV0)∗ ωα (q) = ωα (p) + h3 (V0)ω

α (q)

using the parallel property of Wa and τ (hence ωα). (Here h3 (V0) de-pends on V0 ∈ C1,α

(Γ(NA′

ε/M

))smoothly by similar reason of h1 and


h2. Since ωα is a 3-form, h3 (V0) can have terms of cubic power order of

V0.) Therefore

(expV0)∗ d (iV1ω

α)− d (iV ωα)

= d[(expV0)

∗ (id(exp V0)V+h2(V0)V ωα)− iV ω

α]

= d[(expV0)

∗ id(exp V0)V ωα + (expV0)

∗ ih2(V0)V ωα − iV ω

α]

= d[iV (expV0)

∗ ωα + (expV0)∗ ih2(V0)V ω

α − iV ωα]

= d[iV (ωα + h3 (V0)ω

α) + (expV0)∗ ih2(V0)V ω

α − iV ωα]

= d[iV (h3 (V0)ω

α (q)) + (expV0)∗ ih2(V0)V ω

α (q)].

Using the property (85) of hi (i = 1, 2, 3), we observe that each termin the above last identity in Cα norm smoothly depends on V0 ∈ C1,α(Γ(NA′

ε/M

))and linearly on V ∈ C1,α

(Γ(NA′

ε/M

)). Also when V0 = 0,

we have (expV0)∗ d (iV1ω

α)− d (iV ωα) = 0. Therefore


α)‖Cα(A′ε)≤ C ‖V0‖C1,α(A′

ε)‖V ‖C1,α(A′

ε).

q.e.d.

Remark 25. We also have some point estimates tied to the featurethat τ is a 3-form. It is well known that for any smooth embeddingsϕ : Aε → M and smooth sections V0 of TM |A′

εwith |V0| smaller than

the injectivity radius of M ,∣∣d expϕ V0

∣∣ ≤ C6 (|dϕ|+ |∇V0|) ,where C6 is a constant only depending on (M,g). Hence for the mapexpV0 : TM |A′

ε→M and any section V of TM |A′

ε, at p ∈ A′

ε we have

|(d expV0) (p)V | ≤ C6 (|dϕ|+ |∇V0|) |V (p)| .So for the first term (83), we have

|[(expV0)∗ d (iV1ωα)− d (iV ω

α)]⊗Wα (p)|≤ C7 (|dϕ|+ |∇V0|)3 (|∇V | |V0|+ |V | |∇V0|+ |V | |V0|) (p) ,

where the power 3 is because ωα is a 3-form. For the next row (84), it is a0-th order linear differential operator on V , where V1 = (d exp V0) |Aε(0)V .

By the C2 smoothness of τ = ωα ⊗Wα and

iW dωα (p) = ∇WWα (p) = 0

for any W ∈ Γ(NA′

ε/M

), at q = expp V0 we have

|iV1dωα (q)| ≤ C4 |V0| |V | , |∇V1Wα (q)| ≤ C4 |V0| |V | .

Hence

|((expV0)∗ ⊗ TV0) [iV1dωα ⊗Wα + ωα ⊗∇V1Wα] (p) +B (V0, V )Wα (p)|

≤ C36 (|dϕ|+ |∇V0|)3 · 2C4 |V0| |V | (p) + C5 |V0| |V | (p)


where the power 3 is because ωα is a 3-form. Combining all, we have∣∣F ′ (V0)V − F ′ (0)V∣∣ (p)

≤[C7 (|dϕ| + |∇V0|)3 (|V0| |V |+ |∇V0| |V |+ |V0| |∇V |) + C5 |V0| |V |

](p) .

Because of the cubic power terms, it is difficult to get a similar quadraticestimate in the Lp setting, namely

∥∥F ′ (V0)V − F ′ (0)V∥∥Lp

(

A′ε,NA′

ε/M

)

≤ C ‖V0‖W 1,p(

A′ε,NA′

ε/M

) ‖V ‖W 1,p

(

A′ε,NA′

ε/M

)

.

This is one of the key reasons that we choose the Schauder setting forimplicit function theorem, where the right inverse bound of F ′ (0) ismuch harder to obtain than in the Lp setting. In contrast, the Cauchy-Riemann operator of J-holomorphic curves is more linear in the Lp

setting: it has the quadratic estimate (see Proposition 3.5.3 in [30])∥∥F ′ (V0)V − F ′ (0)V

∥∥Lp(Σ)

≤ C ‖V0‖W 1,p(Σ) ‖V ‖W 1,p(Σ).

4.4. Perturbation argument. To find the zeros of Fε, we are going toapply the following quantitative version of the implicit function theorem(cf. Theorem 15.6 [9] or Proposition A3.4 in [30]).

Theorem 26. Let (X, |·|X) and (Y, |·|Y ) be Banach spaces and F :Br (0) ⊂ X → Y a C1-map, such that

1) (DF (0))−1 is a bounded linear operator with∥∥∥(DF (0))−1 F (0)

∥∥∥ ≤

A and∥∥∥(DF (0))−1

∥∥∥ ≤ B;

2) ‖DF (x)−DF (0)‖ ≤ κ |x|X for all x ∈ Br (0) ;3) 2κAB < 1 and 2A < r.

Then F has a unique zero in B2A (0) .

To apply the above theorem, we define the map

Fε := ε−(

3p+2α

)

Fε : C1,α−(A′ε, NA′

ε/M

)−→ Cα

(A′ε, NA′

ε/M

).

Then Proposition 22 implies that∥∥∥∥(DFε (0)

)−1∥∥∥∥ ≤ ε

3p+2α · Cε−

(

3p+2α

)

= C.

By Proposition 23, for ‖V0‖C1,α−

(

A′ε,NA′ε/M

) ≤ δ0 and any V ∈ C1,α−

(A′ε, NA′

ε/M

),

∥∥∥(DFε (V0)−DFε (0)

)V∥∥∥Cα

(

A′ε,NA′ε/M

)

≤ Cε−(

3p+2α

)

‖V0‖C1,α−

(

A′ε,NA′ε/M

) ‖V ‖C1,α

−

(

A′ε,NA′ε/M

) .


For F (V ) = TV (expV )∗ τ , fixing a volume form on A′ε, we can

regard F (V ) as a section of TM |A′ε . We have

‖F (0)‖Cα(A′ε,TM)

= ‖ϕ∗τ‖Cα(A′ε,TM)

= C

∥∥∥∥(ϕ∗τ) (0, z) +

∫ x1

0

∂

∂x1(ϕ∗τ) (s, z) ds

∥∥∥∥Cα(A′ε,TM)

= C

∥∥∥∥∫ x1

0

∂

∂x1ϕ∗τ (s, z) ds

∥∥∥∥Cα(A′ε,TM)

≤ Cε1−α,(86)

where (ϕ∗τ) (0, z) = 0 is because ϕ (0× Σ) is Jn-holomorphic and thenτ |TA′ε|ϕ(0×Σ)

= 0, and the last inequality is because for

H (x1, z) :=

∫ x1

0

∂

∂x1ϕ∗τ (s, z) ds

where f (s, z) := ∂∂x1

ϕ∗τ (s, z) is smooth on [0, ε]× Σ, we have

‖H‖C0(Aε)≤∫ ε

0‖f‖C0(Aε)

ds = ‖f‖C0(Aε)ε,

[H]zα;Aε ≤∫ ε

0[f ]zα;Aε ds = [f ]zα;Aε ε,

[H]x1α;Aε

≤ C [H]x11;Aε

ε1−α ≤ ‖f‖C0(Aε)ε1−α.

(Here [f ]zα;Aε , [f ]x1α;Aε

are the Schauder components of f in the z and x1directions of Aε.) Note that (86) implies that the tangent space of A′ε isε1−α-close to being associative.

By our construction

Fε (V ) = ∗A′ε ⊥A′ε F (V ) ,

where ∗A′ε ⊥A′ε is a bounded linear operator and TV is a parallel trans-port, we have

∥∥∥Fε (0)∥∥∥Cα

(

Aε,NA′ε/M

) ≤ Cε−(

3p+2α

)

‖F (0)‖Cα

(

Aε,NA′ε/M

)

≤ Cε−(

3p+2α

)

ε1−α = Cε1−

(

3p+3α

)

.

Then we have∥∥∥DFε (0)

−1∥∥∥∥∥∥Fε (0)

∥∥∥Cα(Aε,S)

ε−(

3p+2α

)

≤ Cε1− 6

p−5α

.

Note we have chosen that 0 < 3p + 3α < 1

2 (say α = 1/12 and p > 12)

in the beginning, so 1− 6p − 5α > 0, then

∥∥∥DFε (0)−1∥∥∥∥∥∥Fε (0)

∥∥∥Cα(Aε,S)

ε−(

3p+2α

)

≤ Cε1−

(

3p+3α

)

ε−(

3p+2α

)

= Cε1−6p−5α → 0


as ε→ 0. Theorem 26 then implies that there is a unique Vε ∈ Γ(NA′

ε/M

)

with

(87) ‖Vε‖C1,α(

A′ε,NA′ε/M

) ≤ 2Cε1−

(

3p+3α

)

that solves Fε (Vε) = 0, i.e. Fε (Vε) = 0. Note that (86) and (87) together

imply that the tangent space of Aε (Vε) is ε1−

(

3p+3α

)

-close to being as-sociative. By Proposition 8, for almost associative submanifolds Aε (Vε),

Fε (Vε) = 0 ⇔ F (Vε) = 0,

while the latter means that Aε (Vε) is associative.Finally, we obtain our main result:

Theorem 27. Suppose that M is a G2-manifold and Ct is a oneparameter family of coassociative submanifolds in M . Suppose that theself-dual two form η = dCt/dt|t=0 ∈ Ω2

+ (C) is nonvanishing, then itdefines an almost complex structure J on C0.

For any regular J-holomorphic curve Σ in C0, there is an instantonAε in M which is diffeomorphic to [0, ε]×Σ and ∂Aε ⊂ C0 ∪Cε, for allsufficiently small positive ε.

Finally, we expect that any instanton A in M bounding C0 ∪Ct andwith small volume must arise in the above manner. Namely, we need toprove a ε-regularity result for instantons.

A few remarks are in order: First, counting such small instantons isbasically a problem in four manifold theory because of Bryant’s result[7], which says that the zero section C in Λ2

+ (C) is always a coasso-ciative submanifold for an incomplete G2-metric on its neighborhood,provided that the bundle Λ2

+ (C) is topologically trivial. Second, when ηhas zeros, the above Theorem 27 should still hold true. However, usingthe present approach to prove it would require a good understandingof the Seiberg-Witten theory on any four manifold with a degeneratedsymplectic form as in the Taubes program. But at least for regular Jη-holomorphic curves Σ in C0 that are away from the zero locus of η,the instantons A

′ε in Theorem 27 still exist nearby to Σ ⊂ M , for our

gluing analysis only involves the local geometry of Σ in M . Third, ifwe do not restrict to instantons of small volume, then we have to takeinto account the compactification of the moduli of instantons whichhas not been established yet, e.g. bubbling phenomenon as for pseudo-holomorphic curves, and gluing of instantons of big and small volumessimilar to [32] in the Floer trajectory case. Nevertheless, one would ex-pect that if the volume of At’s is small, then bubbling cannot occur; thusthey would converge to a Jη-holomorphic curve in C0.


5. Appendix: The exponential-like map exp

We construct the exponential-like map exp : S →M satisfying theproperties 1∼3 in Subsection 4.2.

For any section V = (u, v) of the spinor bundle S →Aε, since thebundle S → Aε is a Cartesian product of the spinor bundle SΣ → Σwith the interval [0, ε], we may alternatively regard V (t, z) as a oneparameter family of sections of the bundle SΣ → Σ for the parametert ∈ [0, ε]. Our goal is to obtain the deformation of A′

ε ⊂M from V . Theidea is to deform Aε ⊂ C : = [0, ε]×C using only the u component, thenmap the deformed Aε to M by ϕ, and at last deform it in M by thev component. The coassociative boundary condition is preserved underthe map exp, since we separate the u and v deformations before andafter the map ϕ, respectively. The precise description is in order.

For each fixed t ∈ [0, ε], u (t, z) and v (t, z) are sections of the spinorbundle SΣ → Σ. Using the real vector bundle isomorphism

(id, f) : NΣ/C ⊕ ∧0,1C

(NΣ/C

)≃ NΣ/C ⊕NC/M |Σ,

u (t, z) and f (v (t, z)) are sections of NΣ/C and NC/M |Σ, respectively.For any fixed z ∈ Σ, using the u component, we have a one parameterdeformation expCz u (t, z) in C for 0 ≤ t ≤ ε, where expC is the expo-nential map associated to the induced metric of C in M . Alternatively,we can view the deformation in the product space C : = [0, ε] × C,such that the line [0, ε] × z ⊂ C is deformed to the curve(t, expCz u (t, z)

)|0 ≤ t ≤ ε

⊂ C. Let the curve γu in M be

γu (t, z) := ϕ(t, expCz u (t, z)

), for 0 ≤ t ≤ ε and z ∈ Σ.

It is clear that

γu (t, z) ⊂ Ct := ϕ (t, C)

since expCz u (t, z) ⊂ C.For the v component, for each fixed t ∈ [0, ε], we have f (v (t, z)) ∈

NC/M |Σ (z). Let

Tγu(t,z) : NC/M |γu(0,z) −→ NC/M |γu(t,z)be the parallel transport along the curve γu (s, z) for 0 ≤ s ≤ t withrespect to the connection of NC/M induced from the metric g on M . Wedefine the exponential-like map exp as follows:

exp : S = S+ ⊕ S

− −→ M

V (t, z) = (u (t, z) , v (t, z)) expMγu(t,z)

(Tγu(t,z)f (v (t, z))

)

where expM is the exponential map in M . It follows from our construc-tion that for any V ∈ Cm

− (Aε,S) and x ∈ ∂Aε = 0, ε×Σ, expV satisfiesthe boundary condition expV |∂Aε ⊂ C0 ∪ Cε, because

(expV ) (x) = expMγu(x) (0) = γu (x) ⊂ ϕ (0, ε × C) = C0 ∪ Cε.


By the definition of exp, it is easy to see exp|Aε = ϕ, because Aε is thezero section of S. Hence on base directions of S → Aε, dexp|Aε = dϕ|Aε ,especially

dexp|0×Σ = id : TΣ → TΣ and dexp|0×Σ :∂

∂x1→ n (z) .

On fiber directions, that dexp|0×Σ = (id, f) : NΣ/C ⊕ ∧0,1C

(NΣ/C

)→

NΣ/C ⊕NC/M |Σ follows from d expM (0) = id.

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The Institute of Mathematical SciencesThe Chinese University of Hong Kong

Shatin, Hong Kong

E-mail address: [email protected]

Department of Mathematics & Computer ScienceRutgers University–Newark

Newark, NJ 07102


Department of MathematicsUniversity of MinnesotaMinneapolis, MN 55455

andDepartment of Mathematics

Harvard UniversityCambridge, MA 02138


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